Gradient Flows, Convexity, and Adjoint Orbits
by
Xuhua Liu
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 4, 2012
Keywords: Lie group, Lie algebra, semisimple, reductive, adjoint orbit, gradient ow,
numerical range, Hessian, connectedness, convexity.
Copyright 2012 by Xuhua Liu
Approved by
Tin-Yau Tam, Chair, Professor of Mathematics
Randall R. Holmes, Professor of Mathematics
Huajun Huang, Associate Professor of Mathematics
Abstract
This dissertation studies some matrix results and gives their generalizations in the con-
text of semisimple Lie groups. The adjoint orbit is the primary object in our study.
The dissertation consists of four chapters. Chapter 1 is a brief introduction about the
interplay between matrix theory and Lie theory.
In Chapter 2 we introduce some structure theory of semisimple Lie groups and Lie
algebras. It involves the root space decompositions for complex and real semisimple Lie
algebras, Cartan decomposition and Iwasawa decomposition for real semisimple Lie algebras
and Lie groups. They play signi cant roles in our generalizations.
In Chapter 3 we introduce a famous problem on Hermitian matrices proposed by H.
Weyl in 1912, which has been completely solved. Motivated by a recent paper of Li et al.
[34] we consider a generalized problem in the context of semisimple as well as reductive Lie
groups. We give the gradient ow of a function corresponding to the generalized problem.
This provides a uni ed approach to deriving several results in [34].
Chapter 4 is essentially a brief survey on some generalized numerical ranges associated
with Lie algebras. The classical numerical range of ann ncomplex matrix is the image of the
unit sphere in Cn under the quadratic form. One of the most beautiful properties is that the
numerical range of a matrix is always convex. It is known as the Toeplitz-Hausdor theorem.
We give another proof of the convexity of some generalized numerical range associated with
a compact Lie group. The Toeplitz-Hausdor theorem becomes a special case.
ii
Acknowledgments
This dissertation arose out of my ve years of graduate study and research at Auburn
University, where I have worked with many nice people whose contributions to the completion
of this work deserved special mention. It is a pleasure to convey my gratitude to them all in
this acknowledgment.
First of all, my deepest gratitude goes to the Most High God, Who is my Creator and
Savior. He also gives me knowledge and strength to complete this work: \(Psalm 127:1)
Unless the Lord builds the house, They labor in vain who build it; Unless the Lord guards
the city, The watchman stays awake in vain."
I am grateful in every possible way to Professor Tin-Yau Tam for his supervision, advice,
and guidance as well as giving me extraordinary experiences of research throughout the ve
years. He is an outstanding graduate mentor, and I have been particularly impressed by
his foresight on success for his students. His mathematical intuition exceptionally inspired
and enriched my growth as a student and a researcher. Above all, he provided me steadfast
encouragement and support in various ways, not only in academic but also in personal life.
He is my spiritual mentor as well. I am indebted to him much more than he knows.
I gratefully acknowledge Professor Randall R. Holmes for his supervision and many
e orts and valuable comments on the rst completed manuscript of my dissertation. He
taught me extraordinarily excellent classes on group theory and representation theory, where
he nourished my mathematical maturity that I will bene t from for a lifetime. His rigorous
and meticulous style of mathematical writing has been inspiring me in preparing research
papers. He is among the best teachers in my student life.
iii
Many thanks go to Dr. Huajun Huang for his valuable advice and supervision. I learned
a lot from the Lie group class taught by him. Our discussions in the Lie theory seminar were
interesting and helpful.
My parents deserve special mention for their support and encouragement. My father,
Xuedun Liu, is the person who cultivated my learning character. My mother, Ziying Guo
raised me with her caring love and contributed to my tough character. She would have been
very proud to see the completion of my PhD study.
I thank my wife, Yaqin Xiao, for her unsel sh love since we knew each other. Her
persistent con dence in me was a great source of encouragement, which drove me to move
forward steadfastly. Our lovely children Hannah and Joseph have been inspiring me to work
hard and e ciently.
Finally, I would like to thank everybody who was important to the successful completion
of this dissertation.
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Structure Theory of Semisimple Lie Groups and Lie Algebras . . . . . . . . . . 3
2.1 Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Lie Groups and Their Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Complex Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Real Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Cartan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Root Space Decomposition and Iwasawa Decomposition . . . . . . . . . . . . 13
2.7 Weyl Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Gradient Flows for the Minimum Distance to the Sum of Adjoint Orbits . . . . 17
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Gradient Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Special Case: A2p, B2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Global Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Convexity of Generalized Numerical Ranges Associated with Lie Algebras . . . . 43
4.1 Classical Numerical Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Generalized Numerical Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Generalized Numerical Ranges Associated with Lie Algebras . . . . . . . . . 45
4.3.1 Compact Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
v
4.3.2 Complex Semisimple Case . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.3 Real Semisimple Case . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vi
Chapter 1
Introduction
This dissertation is a study of some matrix results and their generalizations in the
context of semisimple Lie groups. It is universally believed that matrix theory has many
applications in various branches of mathematics and sciences. Matrix theory has a close
relationship with the theory of Lie groups. For example, the general linear groups GLn(C)
and GLn(R) are Lie groups. Roughly speaking, a Lie group is a smooth manifold which
is also a group and in which the group operations are smooth. The tangent space at the
identity of a Lie group has a Lie algebra structure, which captures most information about
the Lie group via the exponential map. The classical Lie groups are matrix groups. This
close connection between matrix theory and Lie theory is bene cial to both elds: matrix
theory provides various results which may lead to new developments of Lie theory and, in
turn, Lie theory often provides uni ed approaches to matrix results and thus inspires deeper
understanding of them.
The main tools in this dissertation are some important decompositions of Lie algebras
and their counterparts for Lie groups. These decompositions are Cartan decomposition and
Iwasawa decomposition which are corresponding to Hermitian decomposition (algebra level),
polar decomposition (group level), and QR decomposition in matrix theory. They exist for
semisimple Lie algebras and Lie groups as well as for reductive Lie algebras and Lie groups.
These decompositions re ect the rich structures of Lie groups. For example, if G is a
connected semisimple Lie group with Lie algebra g and if is a Cartan involution of g, then
g = k p is a Cartan decomposition, where k and p are the +1 and 1 eigenspaces of ,
respectively, and the Killing form B on g induces a positive de nite symmetric bilinear form
B on g de ned by B (X;Y) = B(X; Y). The bilinear form B , together with adjoint
1
orbits in g, enables one to do fruitful analysis on the Lie group G via the exponential map
from g to G.
In matrix theory, any complex n n matrix is an element of gln(C), the Lie algebra
of the general linear group GLn(C) and the Lie bracket is given by [X;Y] = XY YX.
If we de ne a Cartan involution of g by (X) = X for all X 2 g, the corresponding
Cartan decomposition is just the Hermitian decomposition and B is (up to a positive scalar
multiple) the usual inner product given by B (X;Y) = trXY .
The following is the structure of this dissertation. In Chapter 2, we introduce some
structures of semisimple Lie groups and Lie algebras for future reference. They are root
space decompositions for complex and real semisimple Lie algebras, Cartan and Iwasawa
decomposition for real semisimple Lie algebras and Lie groups. In Chapter 3, we consider a
generalization of a famous problem on sum of Hermitian matrices proposed by Weyl in the
context of semisimple as well as reductive Lie groups, where we will derive the gradient ow
and provide a uni ed approach to several results in [34]. Chapter 4 is essentially a brief survey
on some generalized numerical ranges associated with Lie algebras. The classical numerical
range of a complex square matrix is the image of the unit sphere under the quadratic form.
One of the most beautiful properties is that the numerical range of a matrix is always convex.
We give another proof of the convexity of a generalized numerical range associated with a
compact Lie group via a connectedness argument. The main tools are a connectedness result
of Atiyah [1] and a Hessian index result of Duistermaat, Kolk and Varadarajan [16].
2
Chapter 2
Structure Theory of Semisimple Lie Groups and Lie Algebras
In this chapter, we introduce most notations in the dissertation and summarize some
structures of semisimple Lie groups and Lie algebras for later reference. Since a Lie group is
simultaneously a smooth manifold and a group such that the group operations are smooth,
we begin with smooth manifolds, for which our main references are [32] and [50].
2.1 Smooth Manifolds
A topological manifold of dimension n is a second countable Hausdor topological space
of which every point has an open neighborhood that is homeomorphic to an open subset of
Rn. Let M be a topological manifold of dimension n. A chart on M is a pair (U;?), where
U M is open and ? is a homeomorphism of U onto an open subset of Rn. Recall that a
map F : U!V, where U and V are open subsets of Rn and Rm, respectively, is said to be
C1 or smooth if each of the component functions of F has continuous partial derivatives of
all orders. A smooth structure on M is a collection of charts f(U ;? ) : 2Ig such that
(1)
[
2I
U = M,
(2) ? ? 1 is C1 for all ; 2I, and
(3) the collection is maximal with respect to (2).
A topological manifold with a smooth structure is called a smooth manifold, or simply
manifold unless otherwise speci ed. A chart on a manifold is said to be smooth if it is
an element of its smooth structure.
Let M and N be manifolds. A continuous map F : M!N is said to be smooth if for
every p2M, there exist smooth charts (U;?) containing p and (V; ) containing F(p) such
3
that F(U) V and the composite map F ? 1 is C1 from ?(U) to (V). If N = R,
F is called a smooth function on M if for every p2M, there exists a smooth chart (U;?)
containing p such that F ? 1 is C1. Let C1(M) denote the set of all smooth functions on
M. A linear map v : C1(M)!R is called a derivation at p2M if it satis es
v(fg) = f(p)v(g) +g(p)v(f); 8f;g2C1(M):
The set Tp(M) of all derivations of C1(M) at p forms a vector space called the tangent space
to M at p. Elements of Tp(M) are called tangent vectors at p.
Let F : M!N be a smooth map and let p2M. The di erential of F at p is the linear
map dFp : Tp(M)!TF(p)(N) de ned by
dFp(v)(f) = v(f F); 8v2Tp(M);8f2C1(N):
The rank of F at p 2 M is the rank of dFp. The smooth map F is an immersion if
rankF = dimM at every p2M. A submanifold of M is a subset S M endowed with
a manifold topology and a smooth structure, i.e., S is a smooth manifold in its own right,
such that the inclusion map : S!M is an immersion.
The tangent bundle T(M) of M is the disjoint union of the tangent spaces at all points
of M. The projection map : T(M) ! M is de ned by sending each vector in Tp(M)
to p2M. The tangent bundle has a natural topology and smooth structure that make it
into a manifold such that : T(M) !M is a smooth map. A vector eld on M is a map
X : M !T(M) such that Xp := X(p) 2Tp(M) for all p2M. The set of smooth vector
elds on M forms in the obvious way a vector space over R; it is also a module over the
ring C1(M): if X is a vector eld on M and f2C1(M), then Xf2C1(M) is de ned by
Xf(p) = Xp(f). Note that a vector eld X on M is R-linear on C1(M) and satis es
X(f g) = (Xf) g +f Xg; 8f;g2C1(M):
4
In other words, X acts as a derivation of the R-algebra C1(M). In fact, derivations of
C1(M) can be identi ed with smooth vector elds: A function X : C1(M)!C1(M) is a
derivation if and only if it is of the form X(f) = Xf for some smooth vector eld X on M
[32, p.86]. If X and Y are smooth vector elds on M, then X Y : C1(M)!C1(M) need
not be a smooth vector eld in general, but the Lie bracket [X;Y] := X Y Y X always
is. The space of smooth vector elds on a manifold has the structure of a Lie algebra over
R.
2.2 Lie Groups and Their Lie Algebras
A vector space g over a eld F with a product g g!g, denoted by (X;Y)7![X;Y]
and called the Lie bracket of X and Y, is called a Lie algebra over F if the following three
conditions are satis ed:
(1) The Lie bracket is bilinear.
(2) [X;X] = 0 for all X2g.
(3) The Jacobi identity [X;[Y;Z]] + [Y;[Z;X]] + [Z;[X;Y]] = 0 holds for all X;Y;Z2g.
An example of a Lie algebra is the general linear algebra gl(V) consisting of all linear oper-
ators on a vector space V with the Lie bracket de ned by
[X;Y] = XY YX; 8X;Y 2gl(V):
Let g and h be Lie algebras. A linear transformation ? : g!h is called a homomorphism
if
?([X;Y]) = [?(X);?(Y)]; 8X;Y 2g:
It follows from the bilinearity and the Jacobi identity that the linear transformation
ad : g!gl(g)
5
given by adX(Y) = [X;Y] for all X;Y 2 g is a Lie algebra homomorphism, called the
adjoint representation of g. A subspace s of g is called a subalgebra if [X;Y] 2 s for all
X;Y 2s; it is called an ideal if [X;Y]2s for all X2g and Y 2s.
A Lie group G is both a smooth manifold and a group such that the maps m : G G!G
and i : G!G de ned by multiplication and inversion are smooth. For example, the set
of all nonsingular complex matrices forms a Lie group, called the general linear group and
denoted by GLn(C). Any closed subgroup of GLn(C) is a Lie group, called a closed linear
group.
Let G be a Lie group. For each g 2 G, the left translation Lg : G ! G de ned by
Lg(h) = gh is a di eomorphism of G. A smooth vector eld X on G is left-invariant if X is
Lg-related to itself for every g2G, i.e.,
X Lg = dLg X; 8g2G:
If we regard X as a derivation, left-invariance is expressed by
(Xf) Lg = X(f Lg); 8f2C1(G);8g2G:
The space of left-invariant smooth vector elds on G is closed under the Lie bracket, and
is therefore a Lie algebra g, called the Lie algebra of G. The map X 7! Xe is a vector
space isomorphism of g onto Te(G). If Xe;Ye2Te(G), let [Xe;Ye] denote the tangent vector
[X;Y]e. The vector space Te(G), with the composition rule (Xe;Ye)7![Xe;Ye], forms a Lie
algebra which is identi ed with g.
A smooth map ? : G ! H between Lie groups G and H is called a smooth homo-
morphism if it is also a group homomorphism. The di erential d? : g ! h between the
corresponding Lie algebras g and h is a Lie algebra homomorphism, called the derived ho-
momorphism of ?.
6
Let G be a Lie group with Lie algebra g. A one-parameter subgroup of G is a smooth
homomorphism : R!G. It is a consequence of the theorem of existence and uniqueness
of solutions of linear ordinary di erential equations that the map 7!d (0) is a bijection of
the set of one-parameter subgroups of G onto g [23, p.103]. For each X2g, let X be the
one-parameter subgroup corresponding to X and de ne the exponential map exp : g !G
by exp(X) = X(1). It follows that X(t) = exp(tX) and that the one-parameter subgroups
are the maps t7!exptX for X2g. The exponential map for a closed linear group is given
by the matrix exponential function [29, p.76]. An important property of the exponential
map is its naturality: if ? : G!H is a smooth homomorphism, then ? expg = exph d?.
A submanifold H of G is called a Lie subgroup if H is a Lie group with binary operation
being that induced by the binary operation on G. A Lie subgroup of G is called a closed
subgroup if it is a closed subset of G. The following theorem shows a one-to-one correspon-
dence between connected Lie subgroups of a Lie group and subalgebras of its Lie algebra
[23, p.112, p.115, p.118].
Theorem 2.1. Let G be a Lie group with Lie algebra g. If H is a Lie subgroup of G, then the
Lie algebra h of H is a subalgebra of g. Moreover, h =fX2g : exptX2H for all t2Rg.
Each subalgebra of g is the Lie algebra of exactly one connected Lie subgroup of G.
For each g 2G, let Ig be the inner automorphism of G de ned by x7!gxg 1. The
derived homomorphism of Ig, denoted by Adg, is an automorphism of g. We thus have
exp(Ad(g)X) = g(expX)g 1; 8g2G;8X2g:
In the special case that G is a closed linear group, we have Ad (g)X = gXg 1. Since exp has
a smooth inverse in a neighborhood of e2G, if X is small Ad (g)X is smooth as a function
from a neighborhood of e to g. That is, g7!Adg is smooth from a neighborhood of e into
GL(g). Moreover Adg Adh = Ad (gh) because Ig Ih = Igh. Thus the smoothness is
valid everywhere on G. Therefore Ad : G!GL(g) is a smooth homomorphism, called the
7
adjoint representation of G. The derived homomorphism of Ad is the adjoint representation
ad : g!gl(g) of g [29, p.80]. Consequently we have
Ad (expX) = exp(adX); 8X2g:
The group Aut g of all automorphisms of g is a closed subgroup of GL(g), hence is a Lie
subgroup of GL(g). The Lie algebra Der g of Aut g consists of all derivations of g [23,
p.127]. Since ad g is a subalgebra of Der g, it corresponds to a connected subgroup Int g of
Aut g [23, p.127], which is generated by exp(ad g) and called the adjoint group of g. Since
exp(adX) = Ad (expX) for all X 2 g, we have Int g = AdG if G is connected. The Lie
algebra g is said to be compact if G is compact or, equivalently, the adjoint group Int g is
compact.
The symmetric bilinear form B on g de ned by
B(X;Y) = tr (adX adY); 8X;Y 2g
is called the Killing form, which is associative in the sense that
B([X;Y];Z) = B(X;[Y;Z]); 8X;Y;Z2g:
If is an automorphism of g, then
ad ( X) = adX 1
and thus B( X; Y) = B(X;Y). In particular, B is AdG-invariant.
A Lie algebra g is Abelian if [g;g] = 0; it is simple if it is not Abelian and has no nontrivial
ideals; it is solvable if Dkg = 0 for some k, where D0g = g and Dk+1g = [Dkg;Dkg]; it is
nilpotent if Ckg = 0 for some k, where C0g = g and Ck+1g = [Ckg;g]; it is semisimple if its
8
(unique) maximal solvable ideal, called the radical of g and denoted by Rad g, is trivial (or,
equivalently, its Killing form is nondegenerate); it is reductive if its center z(g) = Rad g (or,
equivalently, [g;g] is semisimple). A Lie algebra is semisimple if and only if it is isomorphic
to a direct sum of simple algebras. A Lie group is called semisimple (simple, reductive,
solvable, nilpotent, Abelian) if its Lie algebra is semisimple (simple, reductive, solvable,
nilpotent, Abelian).
2.3 Complex Semisimple Lie Algebras
Let g be a complex semisimple Lie algebra. An element X 2 g is called nilpotent if
adX is a nilpotent endomorphism; it is called semisimple if adX is diagonalizable. Since g
is semisimple, it possesses nonzero subalgebras consisting of semisimple elements, which are
Abelian and are called toral subalgebras of g [27, p.35]. The normalizer of a subalgebra a of
g is
Ng(a) =fX2g : adX(a) ag;
it is the largest subalgebra of g which contains a and in which a is an ideal. A subalgebra h
of g is called a Cartan subalgebra of g if it is self-normalizing, i.e., h = Ng(h), and nilpotent.
The Cartan subalgebras of g are exactly the maximal toral subalgebras of g [27, p.80]. All
Cartan subalgebras of g are conjugate under the adjoint group Int g of inner automorphisms
[27, p.82].
Let h be a Cartan subalgebra of g. Since h is Abelian, ad gh is a commuting family
of semisimple endomorphisms of g, which are thus simultaneously diagonalizable. In other
words, g is the direct sum of the subspaces
g =fX2g : [H;X] = (H)X for all H2hg;
where ranges over the dual h of h. Note that g0 = h because h is self-normalizing. A
nonzero 2h is called a root of g relative to h if g 6= 0. The set of all roots, denoted by
9
, is call the root system of g relative to h. Thus we have the root space decomposition
g = h
M
2
g :
The importance of root space decomposition lies on the fact that characterizes g com-
pletely.
The restriction of the Killing form on h is nondegenerate and is given by
B(H;H0) =
X
2
(H) (H0); 8H;H02h:
Consequently we can identify h with h : each 2 h corresponds a unique H 2 h with
(H) = B(H ;H) for all H2h, and there is a nondegenerate bilinear form h ; i de ned on
h by h ; i= B(H ;H ) for all ; 2h . The following is a collection of some properties
of the root space decomposition [27, p.36{40]:
(1) is nite and spans h .
(2) If ; 2 [f0g and + 6= 0, then B(g ;g ) = 0.
(3) If 2 , then 2 , but no other scalar multiple of is a root.
(4) If 2 , then [g ;g ] is one dimensional, with basis H .
(5) If 2 , then dim g = 1.
(6) If ; 2 , then 2h ; ih ; i 2Z and 2h ; ih ; i 2 .
2.4 Real Forms
Let g be a complex Lie algebra. Then g can be viewed as a real Lie algebra gR, called
the reali cation of g. A real form of g is a subalgebra g0 of gR such that gR = g0 ig0;
in this case, g is called the complexi cation of g0. Let g0 be a real form of g. Each Z 2 g
10
can be uniquely written as Z = X + iY with X;Y 2 g0. A map : g ! g given by
X + iY 7!X iY (X;Y 2g0) is called a conjugation of g with respect to g0. It is easy to
see that
(1) 2 = 1,
(2) ( X) = (X) for all X2g0 and 2C,
(3) (X +Y) = (X) + (Y) for all X;Y 2g0, and
(4) [X;Y] = [ X; Y] for all X;Y 2g0.
Thus is not an automorphism of g, but it is an automorphism of the real algebra gR. On
the other hand, if : g!g satis es the above properties, the set g0 of xed points of is a
real form of g and is the conjugation of g with respect to g0. Hence there is a one-to-one
correspondence between real forms and conjugations of g.
Let B0, B, and BR denote the Killing forms of the Lie algebras g0, g, and gR, respectively.
Then [23, p.180]
B0(X;Y) = B(X;Y); 8X;Y 2g0
BR(X;Y) = 2ReB(X;Y); 8X;Y 2gR:
Consequently g0, g, and gR are all semisimple if any of them is.
Every complex semisimple Lie algebra has a compact real form [23, p.181]. The compact
real forms of complex simple Lie algebra are list in [23, p.516].
2.5 Cartan Decomposition
Let g be a real semisimple Lie algebra, gC its complexi cation, the conjugation of
gC with respect to g. A direct decomposition g = k p of g into a subalgebra k and a
vector subspace p is called a Cartan decomposition if there exists a compact real form u of
11
gC such that (u) u, k = g\u and p = g\iu. If u is any compact real form of gC with a
conjugation , then there exists an automorphism ? of gC such that the compact real form
?(u) is invariant under , which guarantees the existence of a Cartan decomposition of g.
In this case, the involutive automorphism = is called a Cartan involution of g. The
bilinear form B of g de ned by
B (X;Y) = B(X; Y); 8X;Y 2g
is symmetric and strictly positive de nite. The following theorem establishes a one-to-one
correspondence between Cartan decompositions of a real semisimple Lie algebra and its
Cartan involutions [23, p.184] [39, p.144].
Theorem 2.2. Let g be a real semisimple Lie algebra with the direct sum of subspaces
g = k p. The following statements are equivalent.
(1) g = k p is a Cartan decomposition.
(2) The map : X +Y 7!X Y (X2k;Y 2p) is a Cartan involution of g.
(3) The Killing form is negative de nite on k and positive de nite on p, and [k;k] k,
[p;p] k, [k;p] p.
Let g = k p be a Cartan decomposition. It follows that k and p are the +1 and 1
eigenspaces of , respectively, and that k is a maximal compactly embedded subalgebra of g.
Moreover, k and p are orthogonal to each other with respect to both the Killing form B and
the inner product B .
In the special case of g being a complex semisimple Lie algebra, if u is a compact real
form of g, then gR = u iu is a Cartan decomposition [23, p.185].
The group level of Cartan decomposition is summarized below [23, p.252] [29, p.362].
12
Theorem 2.3. Let G be a noncompact semisimple Lie group with Lie algebra g. Let g = k p
be the Cartan decomposition corresponding to a Cartan involution of g. Let K be the
analytic subgroup of G with Lie algebra k. Then
(1) K is connected, closed, and contains the center Z of G. Moreover, K is compact if
and only if Z is nite. In this case, K is a maximal compact subgroup of G.
(2) There exists an involutive, analytic automorphism of G whose xed point set is K
and whose di erential at e is .
(3) The map K p!G given by (k;X)7!k expX is a di eomorphism onto.
For any k 2 K, Adk leaves B invariant because Adk 2 Aut g; Adk also leaves k
invariant because k is the Lie algebra of K and hence Adk leaves invariant the subspace of
g orthogonal to k, which is exactly p. If X2g, write X = Xk +Xp with Xk 2k and Xp 2p
and we see that
Adk( (X)) = Ad (k)Xk Ad (k)Xp = (Ad (k)Xk) + (Ad (k)Xp) = (Ad (k)X);
i.e., Adk commutes with . Hence Adk leaves B invariant as well.
2.6 Root Space Decomposition and Iwasawa Decomposition
Let g be a real semisimple Lie algebra and g = k p a Cartan decomposition with
the corresponding Cartan involution. The bilinear form B endows g with the structure of
a nite-dimensional inner product space. For any X2g, with respect to B , the adjoint of
adX is ad (X) [29, p.360]. If X 2 p, then adX is represented by a symmetric matrix
with respect to an orthonormal basis of g. Thus the elements of p are semisimple with real
eigenvalues. Let a be a maximal Abelian subspace of p. The commutative family ad a is
13
simultaneously diagonalizable. For each real linear functional on a, let
g =fX2g : [H;X] = (H)X for allH2ag:
It is easy to see that (g ) = g and [g ;g ] g + . If 6= 0 and g 6= f0g, then is
called a root of g with respect to a. Let denote the set of all roots. The simultaneously
diagonalization is expressed by g = g0 L 2 g , which is called the root space decomposition
of g with respect to a.
For each root , the set
P =fX2a : (X) = 0g
is a subspace of a of codimension 1. The subspaces P ( 2 ) divide a into several open
convex cones, called Weyl chambers. Fix a Weyl chamber a+ and refer it as the fundamental
Weyl chamber. A root is positive if it is positive on a+. Let + denote the set of all positive
roots. If 2 + and X2g , write X = Xk + Xp with Xk 2k and Xp 2p. Since [k;p] p
and [p;p] k, for any H2a we have (adH)Xk = (H)Xp and (adH)Xp = (H)Xk, which
imply (adH)2Xk = (H)2Xk, (adH)2Xp = (H)2Xp, and (X) = Xk Xp 2 g . For
2 +, de ne
k =fX2k : (adH)2X = (H)2X for all H2ag;
p =fX2p : (adH)2X = (H)2X for all H2ag:
Let m be the centralizer of a in k, i.e.,
m =fX2k : ad (X)H = 0 for all H2ag:
The following result [36, p.107] is helpful in deriving the Hessian of a smooth function on K
(see Lemma 4.5).
14
Lemma 2.4. (1) k = m
X
2 +
k and p = a
X
2 +
p are direct sums whose components
are mutually orthogonal under B ,
(2) g g = k p for all 2 +, and
(3) dim g = dim k = dim p for all 2 +.
The space n = L 2 + g is a subalgebra of g. If X2L 2 + g , then
X = (X + (X)) (X)2k + n:
Since g0 = (g0\k) a, we see that g = k+a+n. Applying we conclude that g = k a n,
which is called Iwasawa decomposition of g [23, p.263] [29, p.373].
The following theorem summarizes the group level of Iwasawa decomposition [29, p.374].
Theorem 2.5. Let G be a semisimple Lie group with Lie algebra g. Let g = k a n be a
Iwasawa decomposition. Let K, A, and N be the analytic subgroups of G with Lie algebras
k, a, and n, respectively. Then G = KAN and the map (k;a;n)7!kan is a di eomorphism
of K A N onto G.
2.7 Weyl Groups
Let the notations be as in Section 2.6. Let m and M be the centralizers of a in k and in
K, respectively, and M0 the normalizer of a in K, i.e.,
m =fX2k : ad (X)H = 0 for all H2ag
M =fk2K : Ad (k)H = H for all H2ag
M0 =fk2K : Ad (k)a ag:
Note that M and M0 are also the centralizer and normalizer of A in K, respectively, and
that they are closed Lie subgroups of K. Moreover, M is a normal subgroup of M0, and
15
the quotient group M0=M is nite because M and M0 have the same Lie algebra m [23,
p.284]. The nite group W(G;A) = M0=M is called the (analytically de ned) Weyl group
of G relative to A. For w = mwM 2 W(G;A), the linear map Ad (mw) : a ! a does
not depend on the choice of mw 2M0 representing w. It follows that w 7! Ad (mw) is a
faithful representation of W(G;A) on a. Thus we may regard w2W(G;A) as the linear map
Ad (mw) : a!a and W(G;A) as a group of linear operators on a. The Weyl group W(G;A)
also acts on a by w = w 1 for all 2a . The Killing form B is nondegenerate on a,
and thus it induces an isomorphism of a and a by 7!H such that
(H) = B(H ;H); 8 2a ;8H2a
This isomorphism induces an action of W(G;A) on a as follows. If we denote Hw = w H
for all 2a , then for all H2a
(w )(H) = B(Hw ;H) = B(w H ;H) = B(Ad (mw)H ;H)
= B(H ;Ad (mw) 1H) = (Ad (mw) 1H)
= ( Ad (mw) 1)(H)
So the Weyl group W(G;A) acts on a by w = Ad (mw) 1 := w 1.
For each root , the re ection s about the hyperplane P =fX2a : (X) = 0g, with
respect to the Killing form B, is a linear map on a given by
s (H) = H 2 (H) (H
)
H ; 8H2a;
where H is the element of a representing , i.e., (H) = B(H ;H) for all H 2 a. The
group W(g;a) generated by fs : 2 g is called the (algebraically de ned) Weyl group
of g relative to a. When viewed as groups of linear operators on a, the two Weyl groups
W(G;A) and W(g;a) coincide [29, p.383].
16
Chapter 3
Gradient Flows for the Minimum Distance to the Sum of Adjoint Orbits
This chapter introduces a famous problem on the sum of Hermitian matrices proposed
by Weyl. Motivated by a recent paper of Li et al. [34], we study Weyl?s problem in the
context of semisimple as well as reductive Lie groups and give the gradient ow of a function
corresponding to a generalized problem. This provides a uni ed approach to several results
in [34].
3.1 Introduction
Given the eigenvalues of two n n Hermitian matrices A and B, a famous problem of
Weyl [52] is to give a complete description of the eigenvalues of C = A+B. The problem is
completely solved and one may see [14, 17, 31] and their references for historical development.
Its extension to compact Lie groups is given in [41, Theorem 9.3] and in particular the
determination of singular values of the sum of two rectangular complex matrices is given in
[41, p.447{450]. The solution of Weyl?s problem and the result in [41] are not easy to be
used as a checking tool for concrete matrices. For example, 10 10 Hermitian matrices yield
too many inequalities [34], according to the Littlewood-Richardson rule. If we denote by
S(H) the unitary similarity orbit of a Hermitian matrix H, then Weyl?s problem is to nd
necessary and su cient conditions for
S(C) S(A) +S(B) :=fUAU 1 +VBV 1 : U;V 2U(n)g
17
in terms of the eigenvalues of A;B and C. The set inclusion is equivalent to
min
U;V2U(n)
kUAU 1 +VBV 1 CkF = 0;
where k kF is the Frobenius norm on Cn n.
Given complex matrices A0, A1;:::;AN, Li et al. [34] studied the more general prob-
lem of nding the least squares approximation of A0 by the sum of matrices from orbits
S(A1);:::;S(AN), i.e.,
minfkX1 + +XN A0kF : (X1;:::;XN)2S(A1) S(AN)g; (3.1)
where the orbits S(Ai), i = 1;:::;N, are induced by some equivalence class on matrices such
as unitary similarity, unitary equivalence, and unitary congruence. In each case, by Fr echet
di erentiation they derived the gradient ow of a corresponding smooth function, which was
then used to design an algorithm to solve the respective optimization problem.
Motivated by the results in [34], we ask whether there might be a uni ed approach for
studying these problems. The purpose of this chapter is to present such an approach in the
context of semisimple as well as reductive Lie groups. Examples are given to illustrate our
results and their relations to those in [34].
3.2 Formulation
Let G be a connected semisimple Lie group with Lie algebra g. Let g = k p be the
Cartan decomposition corresponding to a Cartan involution of g, where k and p are the
+1 and 1 eigenspaces of , respectively. The Killing form B of g induces a positive de nite
symmetric bilinear form B on g given by B (X;Y) = B(X; Y) for all X;Y 2 g. Note
that B j(k k) = B and B j(p p) = B, and that k and p are orthogonal under both B and B
[29, p.359]. Given X2g, we have the unique decomposition X = Xk +Xp with Xk 2k and
Xp 2p. Let K be the analytic subgroup of G with Lie algebra k. Note that K is compact
18
if and only if G has nite center [29, p.362]. We remark that simple classical groups have
compact K [23, p.446{455]. Though K may not be compact in general, AdGK (or simply
AdK) is compact in the adjoint group Int g = AdG and each Adk2 Aut g is orthogonal
with respect to B . Both k and p are AdK-invariant, i.e., Ad (k)k = k and Ad (k)p = p for
all k2K.
We cast Problem (3.1) in the context of semisimple Lie groups as follows. Given
A0;A1;:::;AN 2g (not necessarily in p), nd
min
ki2K
NX
i=1
Ad (ki)Ai A0
; (3.2)
where the norm k k is induced by B , i.e., kXk2 = B (X;X) for all X 2 g. In other
words, we want to nd the (minimum) distance between A0 and the sum of the orbits
Ad (K)Ai, i = 1;:::;N. The sum Ad (K)A1 + + Ad (K)AN is a union of orbits since it
is invariant under AdK. The minimum is justi ed since AdK is compact. Problem (3.2) is
very di cult in view of the nontrivial solution to Weyl?s problem, which is corresponding to
sln(C) = su(n) isu(n) with A;B;C2p and p = isu(n) consisting of Hermitian matrices.
The solution of O?Shea and Sjmaar [41, Theorem 9.3] is very involved, which is essentially
corresponding to a complex semisimple Lie algebra g = k ik, i.e., p = ik. Both are restricted
to A0;A1; AN 2p. Our goal is not to solve Problem (3.2). Instead we provide its gradient
ow and related results. Since the general case does not di er much from the N = 2 case,
our focus will be on this simpler case.
In Section 3.3, we derive the gradient ow of a smooth function associated to Problem
(3.2). In Section 3.4, we show that several results in [34] can be recovered from our general
results. In Section 3.5, we consider the special case when N = 2 and A1 2p and A2 2k, for
which the minimum and the maximum are given. Finally, some remarks are made for local
and global extrema in Section 3.6.
19
3.3 Gradient Flow
Let the notations be as in Section 3.2. We rst de ne the gradient ow of a smooth
function on the analytic Lie subgroup K of G. It has a bi-invariant Riemannian structure [23,
p.47] induced by the unique bi-invariant Riemannian structure Q on G such that Qe = B
[23, p.148 #5]. More precisely, for each k2K, the right translation Rk : K !K de ned
by Rk(h) = hk is a di eomorphism. Its derivative at the point h 2 K is denoted by
dRk : Th(K)!Thk(K), where Th(K) denotes the tangent space of K at h. The Riemannian
structure on K is given by
hU;Vik := Q(U;V) = B (dR 1k (U);dR 1k (V)) = B(dR 1k (U);dR 1k (V))
for all U;V 2Tk(K). Note that this structure is bi-invariant since
hU;Vik = B(Ad (k 1)dR 1k (U);Ad (k 1)dR 1k (V))
= B(dL 1k (U);dL 1k (V)):
We simply write hU;Vi for hU;Vik if there is no danger of confusion. With respect to this
Riemannian structure, if ? : K!R is a smooth function, the gradient r?k 2Tk(K) of ?
at k is given by
d?k(V) =hr?k;Vik; V 2Tk(K): (3.3)
Sinceh ; ik is nondegenerate,r?k is well de ned and thus we have the gradient vector eld
r? : K!Tk(K). So the gradient ow of ? becomes
dk
dt = r?k: (3.4)
20
Now we return to Problem (3.2) and focus on the simpler case when N = 2, i.e.,
min
k;h2K
kAd (k)A+ Ad (h)B Ck; A;B;C2g: (3.5)
The general case is similar. Note that for any k;h2K
kAd (k)A+ Ad (h)B Ck2
= kAd (k)Ak2 +kAd (h)Bk2 +kCk2 + 2B (Ad (k)A;Ad (h)B)
2B (Ad (k)A;C) 2B (Ad (h)B;C)
= kAk2 +kBk2 +kCk2 + 2B (Ad (k)A;Ad (h)B)
2B (Ad (k)A;C) 2B (Ad (h)B;C):
Thus Problem (3.5) is equivalent to the problem of nding
min
k;h2K
f(k;h); (3.6)
where the smooth function f : K K!R is de ned by
f(k;h) := B (Ad (k)A;Ad (h)B) B (Ad (k)A;C) B (Ad (h)B;C): (3.7)
Apply the previous discussion on K K. The tangent space T(k;h)(K K) of K K at the
point (k;h) is endowed with a bi-invariant Riemannian structure given by
hU;Vi= B(g;g)(dR 1(k;h)U;dR 1(k;h)V); U;V 2T(k;h)(K K); (3.8)
where the Killing form B(g;g) is naturally induced by B. With respect to this structure, the
gradient rf(k;h) of f at the point (k;h) is given by
df(k;h)(V) =hrf(k;h);Vi; V 2T(k;h)(K K): (3.9)
21
The gradient ow of (3.7) becomes
d(k;h)
dt = rf(k;h): (3.10)
Theorem 3.1. The gradient ow of f de ned in (3.7) is given by
d(k;h)
dt = ( dRk(t)[ (C Ad (h)B);Ad (k)A]k; dRh(t)[ (C Ad (k)A);Ad (h)B]k); (3.11)
or equivalently,
dk
dt = dRk(t)[ (C Ad (h)B);Ad (k)A]k; (3.12)
dh
dt = dRh(t)[ (C Ad (k)A);Ad (h)B]k: (3.13)
When A;B;C2p, the gradient ow becomes
d(k;h)
dt = (dRk(t)[C Ad (h)B;Ad (k)A];dRh(t)[C Ad (k)A;Ad (h)B]): (3.14)
When A;B;C2k, the gradient ow becomes
d(k;h)
dt = ( dRk(t)[C Ad (h)B;Ad (k)A]; dRh(t)[C Ad (k)A;Ad (h)B]): (3.15)
Proof. Each element V 2T(k;h)(K K) is of the form V = dR(k;h)(X;Y) for some unique
(X;Y) 2 (k;k). The curve et(X;Y)(k;h) passes through (k;h) with tangent vector V. Note
that B([X;Y];Z) = B(X;[Y;Z]) for all X;Y;Z 2 g [23, p.131], and that k and p are
22
orthogonal under B. We have
df(k;h)(V) = ddt
t=0
f(et(X;Y)(k;h)) = ddt
t=0
f((etXk;etYh))
= ddt
t=0
fB (Ad (etXk)A;Ad (etYh)B) B (Ad (etXk)A;C)
B (Ad (etYh)B;C)g
= B([X;Ad (k)A]; Ad (h)B) B([Y;Ad (h)B]; Ad (k)A)
+B([X;Ad (k)A]; C) +B([Y;Ad (h)B]; C)
= B([Ad (k)A; Ad (h)B];X) B([Ad (h)B; Ad (k)A];Y)
B([ C;Ad (k)A];X) B([ C;Ad (h)B];Y)
= B(g;g)([ (C Ad (h)B);Ad (k)A];[ (C Ad (k)A);Ad (h)B];X Y)
=hdR(k;h)([ (C Ad (h)B);Ad (k)A]k;[ (C Ad (k)A);Ad (h)B]k);Vi by (3.8)
Thus
rf(k;h) = (dRk[ (C Ad (h)B);Ad (k)A]k;dRh[ (C Ad (k)A);Ad (h)B]k) (3.16)
and the gradient ow takes the desired form (3.11). The results for A;B;C2p and A;B;C2
k follow from the facts that p and k are AdK-invariant, and that [p;p] k and [k;k] k [29,
p.359], and that k and p are the +1 and 1 eigenspaces of , respectively.
Similarly we have the following result for the general case.
Theorem 3.2. The gradient ow of the smooth function associated to Problem (3.2) is given
by the following system of di erential equations
dki
dt = dRki(t)[ (A0
X
j6=i
Ad (kj)Aj);Ad (ki)Ai]k; i = 1;:::;N: (3.17)
23
Remark 3.3. The above results in this section are also true for reductive Lie groups (proofs
are skipped), which are members of the Harish-Chandra class [29, p.446]. More precisely, a
reductive Lie group is a 4-tuple (G;K; ;B), where G is a Lie group, K is a compact subgroup
of G, is a Lie algebra involution of the Lie algebra g of G, and B is an AdG-invariant,
-invariant, nondegenerate symmetric bilinear form on g such that
(1) g is reductive, i.e., g = z [g;g], where z is the center of g and [g;g] is semisimple,
(2) g = k p, where k and p are the +1 and 1 eigenspaces of , respectively, and k is also
the Lie algebra of K,
(3) k and p are orthogonal with respect to B, and B is negative de nite on k and positive
de nite on p,
(4) the map K exp p!G given by multiplication is a di eomorphism onto,
(5) for every g2G, the automorphism Adg of g, extended to the complexi cation gC of
g, is contained in Int gC, and
(6) the semisimple connected subgroup Gss of G with Lie algebra [g;g] has nite center.
Example 3.4. Let G be a semisimple Lie group with nite center, let B be the Killing form
on the Lie algebra g of G, let : g!g be a Cartan involution, let g = k p be the Cartan
decomposition with respect to , and let K be the analytic subgroup of G with Lie algebra
k. Then (G;K; ;B) is a reductive Lie group.
Remark 3.5. The gradient ow method has been used for other studies. For instance,
a variety of algorithms in numerical analysis can be approached through gradient ows
on adjoint orbits associated with semisimple Lie groups [9]. The recent review [43] gives
a comprehensive account on the foundations of gradient ows on Riemannian manifolds
including new applications to quantum control. See also [3, 8, 12, 24, 47]. See [53] for some
study involving the sum of adjoint orbits associated with a compact connected Lie group.
24
3.4 Examples
In this section, we show by examples that several results in [34] can be recovered from
our general results. Fr echet derivative is the main tool in [34] to derive various gradient
ows. Our approach in Theorem 3.1 is intrinsic, i.e., no ambient space is required as in
Fr echet di erentiation.
Example 3.6. Consider the reductive group G = GLn(C) with Lie algebra g = gln(C)
and (X) = X . Then g = k p is just the Hermitian decomposition with k = u(n)
consisting of skew-Hermitian matrices and p = iu(n) consisting of Hermitian matrices. Let
the nondegenerate symmetric bilinear form be de ned by B(X;Y) = Re trXY for all X;Y 2
g. Now B (X;Y) = Re trXY and the norm induced by B is the Frobenius norm. Problem
(3.5) is then
min
U;V2U(n)
kUAU +VBV CkF: (3.18)
By Remark 3.3 (we cannot directly apply Theorem 3.1 since gln(C) is not semisimple) the
associated gradient ow becomes
dU
dt = dRU[ (C Ad (V)B);Ad (U)A]k = [(C
VB V );UAU ]kU;
dV
dt = dRV [ (C Ad (U)A);Ad (V)B]k = [(C
UA U );VBV ]kV;
which are exactly the formulae in [34, Section 2.4].
Example 3.7. Consider the group G = Up;q [29, p.115], whose Lie algebra is
up;q =
8
><
>:
0
B@X1 Y
Y X2
1
CA : X
1 2u(p);X2 2u(q);Y 2Cp q
9
>=
>;:
25
Let (X) = Ip;qXIp;q where Ip;q = ( Ip) Iq. Then we have
k =
8
><
>:
0
B@X1 0
0 X2
1
CA : X
1 2u(p);X2 2u(q)
9
>=
>;;
p =
8
><
>:
0
B@ 0 Y
Y 0
1
CA : Y 2C
p q
9
>=
>;;
K = U(p) U(q) =
8>
<
>:
0
B@U 0
0 V
1
CA : U2U(p);V 2U(q)
9
>=
>;:
The group action of K on p is given by
^A :=
0
B@ 0 A
A 0
1
CA7!
0
B@U 0
0 V
1
CA
0
B@ 0 A
A 0
1
CA
0
B@U 0
0 V
1
CA =
0
B@ 0 UAV
(UAV) 0
1
CA;
under which the orbit of ^A2p is the set
Ad (K) ^A =
8
><
>:
0
B@ 0 UAV
(UAV) 0
1
CA : U2U(p);V 2U(q)
9
>=
>;:
We set B(X;Y) := Re trXY. For ^A; ^B; ^C2p, the minimization problem
min
k;h2K
kAd (k) ^A+ Ad (h) ^B ^Ck (3.19)
is equivalent to
min
U;X2U(p);V;Y2U(q)
kUAV +XBY CkF; (3.20)
which is studied in [34, Section 3.1]. So we want to minimize the function
f(U;V;X;Y) := Re tr (UAV(XBY C) XBYC ):
26
Then from (3.14) the associated gradient ow to the minimization problem (3.19) becomes
dk
dt = dRk(t)[
^C Ad (h) ^B;Ad (k) ^A]; (3.21)
dh
dt = dRh(t)[
^C Ad (k) ^A;Ad (h) ^B]: (3.22)
Set k = U V ;h = X Y 2K. Then (3.21) and (3.22) become
dU
dt = 2f(UAV)(C XBY)
gu(p)U;
dV
dt = 2f(UAV)
(C XBY)gu(q)V ;
dX
dt = 2f(XBY)(C UAV)
gu(p)X;
dY
dt = 2f(XBY)
(C UAV)gu(q)Y ;
which match the formulae in [34, Section 3.1].
Example 3.8. Consider the simple Lie algebra g = spn(R) and let (A) = A> for all
A2g. Then we have
spn(R) =
8
><
>:
0
B@A1 A2
A3 A>1
1
CA : A>
2 = A2; A
>
3 = A3; A1;A2;A3 2Rn n
9
>=
>;
K =
8>
<
>:
0
B@ U1 U2
U2 U1
1
CA : U>
1 U1 +U
>
2 U2 = I; U
>
1 U2 = U
>
2 U1; U1;U2 2Rn n
9
>=
>;
k =
8>
<
>:
0
B@ A1 A2
A2 A1
1
CA : A>
1 = A1; A
>
2 = A2; A1;A2 2Rn n
9
>=
>;
p =
8
><
>:
0
B@A1 A2
A2 A1
1
CA : A>
1 = A1; A
>
2 = A2; A1;A2 2Rn n
9
>=
>;:
27
As in [44], we identify K with U(n) through the map : K!U(n) de ned by
0
B@ U1 U2
U2 U1
1
CA = U
1 +iU2:
The map preserves matrix multiplication as well as addition. We identify k with u(n) in
the same way. Let S denote the space of n n complex symmetric matrices. We identify p
with S via the map : p!S de ned by
0
B@A1 A2
A2 A1
1
CA = A
2 +iA1:
Note that U 1 = U> for all U2K and that for all A2p,
(Ad (U)A) =
2
64
0
B@ U1 U2
U2 U1
1
CA
0
B@A1 A2
A2 A1
1
CA
0
B@ U1 U2
U2 U1
1
CA
13
75
= (U1 +iU2)(A2 +iA1)(U1 +iU2)>:
Hence with these identi cations, the adjoint action of K on p corresponds to the map A7!
UAU> for A2S and U2U(n).
For A;B;C2S, the minimization problem [34, Section 4.1]
min
U;V2U(n)
kUAU> +VBV> Ck
28
corresponds to Problem (3.5) with g = spn(R) and A;B;C 2 p. The associated gradient
ow is given by (3.14):
dU
dt = [Ad (U)A;C Ad (V)B]U
= (UAU> ~C ~CUAU>)U; (with ~C = C VBV>)
dV
dt = [Ad (V)B;C Ad (U)A]V
= (VBV> ~C ~CVBV>)V; (with ~C = C UAU>):
Remark 3.9. The results in [34, Section 4.1] are for all A;B;C2Cn n. When A;B;C2S,
our results in Example 3.8 match [34].
3.5 Special Case: A2p, B2k
We now consider the special case of Problem (3.5) when A2p and B2k, or vice versa.
The consideration is mainly motivated by [34, Section 2.3] in which the reductive gln(C) is
studied. We rst consider the semisimple case and then make a remark on the reductive
case. Let a be a maximal Abelian subspace of p. Let Wa = M0=M be the Weyl group of G
relative to a, where
M =fk2K : Ad (k)H = H for all H2ag;
M0 =fk2K : Ad (k)H2a for all H2ag:
Since M0 and M have the same Lie algebra [23, p.284], Wa is a nite group. For each
w = kwM2Wa, Adkw : a!a is independent of the representative kw2M0 so that we may
regard Wa as a subgroup of GL(a). The root space decomposition of g with respect to a is
g = g0
X
2
g ;
29
where is the root system and
g =fX2g : adH(X) = (H)X for all H2ag
for 2a . The hyperplanes P =fH2a : (H) = 0gfor 2 divide a into nitely many
open convex cones, which are called Weyl chambers. The Weyl group acts transitively on
the Weyl chambers. Fix a Weyl chamber a+ and refer it as the fundamental Weyl chamber,
whose opposite Weyl chamber is then a+. A root is called positive if it is positive on
a+, and a positive root is called simple if it is not the sum of two positive roots. The set of
simple roots is denoted by . For each root the re ection s about the hyperplane P in
a, with respect to B , is a linear map given by
s (H) = H (H) (H
)
H ; for all H2a;
where H 2 a is the element representing , i.e., (H) = B (H;H ) for all H 2 a. It is
known that the Weyl group Wa is generated by the simple re ections s for 2 [29,
Proposition 2.62, Theorem 6.57]. For each w2Wa, de ne the length of w to be the smallest
integer l such that w can be expressed as a product of l simple re ections. There is a unique
element of maximal length [28, Section 1.8], which we denote by !a and call the longest
element of Wa. This element is also uniquely characterized as the element in Wa that sends
the fundamental Weyl chamber a+ to a+. So it is also known as the opposition element [42,
p.88]. For example, when g = sln(C) and a consists of diagonal matrices in p = isu(n), let
a+ be the set of diagonal matrices with non-increasing diagonal entries of zero trace. Then
we have
!a(diag (a1;a2;:::;an)) = diag (an;:::;a2;a1)2 a+:
There is another Weyl group associated with K that is di erent from the Weyl group
Wa of (g;a). By the proof of [30, Proposition 2.3], we have K = K0Z, where K0 is compact
30
semisimple and Z is the center of K. Let T0 be any maximal Abelian subgroup of K0 and
t0 its Lie algebra. The Weyl group of (K0;T0), de ned to be the quotient of the normalizer
of T0 in K modulo T0, acts by automorphisms of T0, hence by invertible operators on t0
and the maximal Abelian subalgebra t = t0 z of k, where z is the Lie algebra of Z. We
therefore de ne in this way the Weyl group Wt for a xed maximal Abelian subalgebra t of k.
A fundamental Weyl chamber (t0)+ in t0 gives a corresponding fundamental Weyl chamber
t+ = (t0)+ z in t. The longest element wt of Wt is the one that sends the fundamental Weyl
chamber t+ to t+.
Let the notations be as above. Note that p = Ad (K)a [29, p.378] and the Weyl groups
de ned above can be viewed as subgroups of AdK. Since A;C C 2 p, there exist
k0;u2K such that Ad (k0)A2a+ and Ad (u)(C C)=22a+. Since B;C + C2k, there
exist h0;v2K such that Ad (h0)B2t+ and Ad (v)(C + C)=22t+.
Lemma 3.10. Let X;Y 2 a (respectively, t). Suppose that !1 X and Y are in the same
Weyl chamber and that !2 X and Y are in opposite Weyl chambers. Then
kY !1 Xk kY ! Xk kY !2 Xk
for all !2Wa (respectively, Wt).
Proof. Note that Wa and Wt can be viewed as subgroups of AdK. The Killing form is
AdK-invariant, so it is both Wa-invariant and Wt-invariant. Thus the B -induced norm
k k, which is convex, is also both Wa-invariant and Wt-invariant. The lemma then follows
from [25, Corollary 3.10, Proposition 2.8].
For the special case when A 2 p and B 2 k, the minimization problem (3.5) has a
solution. The following result extends [34, Theorem 2.1].
Theorem 3.11. Let A2p, B2k, and C2g. Let k0;u2K such that Ad (k0)A2a+ and
Ad (u)(C C)=22a+, and let h0;v2K such that Ad (h0)B2t+ and Ad (v)(C+ C)=22t+.
31
Then
min
k;h2K
kAd (k)A+ Ad (h)B Ck2
= kAd (u 1k0)A+ Ad (v 1h0)B Ck2 (3.23)
= kAd (u 1k0)A (C C)=2k2
+kAd (v 1h0)B (C + C)=2k2
and
max
k;h2K
kAd (k)A+ Ad (h)B Ck2
= kAd (u 1)! 1a Ad (k0)A+ Ad (v 1)! 1t Ad (h0)B Ck2 (3.24)
= kAd (u 1)! 1a Ad (k0)A (C C)=2k2 +
kAd (v 1)! 1t Ad (h0)B (C + C)=2k2;
where !a and !t are the longest elements of the Weyl groups Wa and Wt, respectively.
Proof. Since k and p are orthogonal under B and k and p are invariant under AdK, we have
kAd (k)A+ Ad (h)B Ck2
= k(Ad (k)A (C C)=2) + (Ad (h)B (C + C)=2)k2
= kAd (k)A (C C)=2k2 +kAd (h)B (C + C)=2k2:
So
min
k;h2K
kAd (k)A+ Ad (h)B Ck2
= min
k2K
kAd (k)A (C C)=2k2 + min
h2K
kAd (h)B (C + C)=2k2
32
and
max
k;h2K
kAd (k)A+ Ad (h)B Ck2
= max
k2K
kAd (k)A (C C)=2k2 + max
h2K
kAd (h)B (C + C)=2k2:
Noting that Wa AdK and Wt AdK and that the longest Weyl group element maps a
Weyl chamber to its opposite chamber, the result then follows from Lemma 3.10.
Example 3.12. We would like to use sln(R) = k p with n = 2m 2 to illustrate Theorem
3.11. Here k = so(n) and p is the space of n n real symmetric matrices of zero trace. Thus
K = SO(n). We may choose [7, p.219]
t =
8>
<
>:
0
B@ 0 1
1 0
1
CA
0
B@ 0 m
m 0
1
CA :
1;:::; m2R
9
>=
>;
t+ =
8
><
>:
0
B@ 0 1
1 0
1
CA
0
B@ 0 m
m 0
1
CA :
1 m 1 j mj
9
>=
>;
a = fdiag ( 1;:::; n) : 1;:::; n2R;
nX
i=1
ai = 0g
a+ = fdiag ( 1;:::; n) : 1 n;
nX
i=1
ai = 0g:
The Weyl group Wa acts as the symmetric group Sn on a, i.e.,
diag ( 1;:::; n)!diag ( (1);:::; (n)); 2Sn
and the Weyl group Wt acts on t in the following way
0
B@ 0 1
1 0
1
CA
0
B@ 0 m
m 0
1
CA!
0
B@ 0 (1)
(1) 0
1
CA
0
B@ 0 (m)
(m) 0
1
CA
33
in which the total number of sign changes on ?s is even and 2Sm.
Suppose C 2 sln(R). According to the Hermitian decomposition and the spectral de-
composition of real symmetric and skew symmetric matrices,
C = U
2
64
0
B@ 0 g1
g1 0
1
CA
0
B@ 0 gm
gm 0
1
CA
3
75U 1 +Vdiag (f
1;:::;fn)V 1 (3.25)
for some U;V 2SO(n), g1 gm 1 jgmj, and f1 fn.
Let A2p and B2k. Then
A = Zdiag (a1;:::;an)Z 1 (3.26)
and
B = W
2
64
0
B@ 0 b1
b1 0
1
CA
0
B@ 0 bm
bm 0
1
CA
3
75W 1 (3.27)
for some Z;W 2 SO(n), b1 bm 1 jbmj, and a1 an. The Killing form of
sln(R) [23, p.180, p.186] is B(X;Y) = 2ntrXY. Since (X) = X>,
B (X;Y) = B(X; Y) = 2ntrXY>
and
kXk2 = 2ntrXX>;
a scalar multiple of the Frobenius norm. By Theorem 3.11
min
k;h2SO(n)
kkAk 1 +hBh 1 Ck2 = 2n
nX
j=1
jfj ajj2 + 2
mX
j=1
jgj bjj2
!
: (3.28)
Note that the scalar multiple 2n is not in [34, Theorem 2.1] because the Frobenius norm is
used there.
34
The longest element of Wa sends
diag ( 1;:::; n)!diag ( n;:::; 1)
and the longest element [42, p.88] of Wt sends
0
B@ 0 1
1 0
1
CA
0
B@ 0 m
m 0
1
CA
!
8
>>>
>>>
>>>>
<
>>>
>>>
>>>
>:
0
BB
@
0 1
1 0
1
CC
A
0
BB
@
0 m
m 0
1
CC
A if m is even
0
BB
@
0 1
1 0
1
CC
A
0
BB
@
0 m 1
m 1 0
1
CC
A
0
BB
@
0 m
m 0
1
CC
A if m is odd
So
max
k;h2SO(n)
kkAk 1 +hBh 1 Ck2
=
8
>>>
>>>>
>>>
<
>>>
>>>
>>>
>:
2n(Pnj=1jfj an j+1j2
+2Pmj=1jgj +bjj2) if m is even
2n(Pnj=1jfj an j+1j2
+2(Pm 1j=1 jgj +bjj2 +jgm bmj2)) if m is odd
(3.29)
However when we view C2sln(C) = su(n) isu(n),
C = iU1diag (g1;:::;gm 1;jgmj; jgmj; gm 1;:::; g1)U 11
+Vdiag (f1;:::;fn)V 1
35
for someU1 2SU(n). Similarly viewA2isu(n) andB2su(n) so thatA = Zdiag (a1;:::;an)Z 1
as before and
B = iW1diag (b1;:::;bm 1;jbmj; jbmj; bm 1;:::; b1)W 11
for some W1 2SU(n). By Theorem 3.11 or [34, Theorem 2.1],
min
k;h2SU(n)
kkAk 1 +hBh 1 Ck2
= 2n
nX
j=1
jfj ajj2 + 2(
m 1X
j=1
jgj bjj2 +jjgmj jbmjj2)
!
: (3.30)
Clearly (3.30) is smaller than or equal to (3.28) because of the triangle inequality jjgmj
jbmjj jgm bmj. Indeed SO(n) SU(n) is the underlying reason.
The two Weyl groups Wa and Wt for sln(C) are equal to Sn. So by Theorem 3.11 or [34,
Theorem 2.1],
max
k;h2SU(n)
kkAk 1 +hBh 1 Ck2
= 2n(
nX
j=1
jfj an j+1j2 + 2(
m 1X
j=1
jgj +bjj2 + (jgmj+jbmj)2)) (3.31)
which is independent of the parity of m.
In conclusion, given C 2 g = k p gC, A 2 p and B 2 k, where gC = u iu
(u = k ip so that A2iu and B2u) is the complexi cation of g, it may not be true that
the corresponding extrema given in Theorem 3.11 are the same for g and gC.
Theorem 3.13. Let C 2 Rn n, A 2 Rn n be symmetric with eigenvalues a1
an and B 2 Rn n be skew symmetric with eigenvalues ib1;:::; ibm when n = 2m and
ib1;:::; ibm;0 when n = 2m + 1. Let A;B;C have decompositions (3.26), (3.27) and
(3.25) respectively, with f1 fn and
1. b1 bm 1 jbmj and g1 gm 1 jgmj when n = 2m
36
2. b1 bm 0 and g1 gm 0 when n = 2m+ 1.
Let k k be the Frobenius norm on Rn n. Then
fkkAk 1 +hBh 1 Ck2 : k;h2SO(n)g= [?0;L0]:
1. If n = 2m, then
?0 =
nX
j=1
jfj ajj2 + 2
mX
j=1
jgj bjj2 (3.32)
and
L0 =
8
>>>
>>>
<
>>>
>>>
:
Pn
j=1jfj an j+1j
2 + 2Pm
j=1jgj +bjj
2 if m even
Pn
j=1jfj an j+1j
2 + 2(Pm 1
j=1 jgj +bjj
2
+jgm bmj2) if m odd
(3.33)
2. If n = 2m+ 1, then
?0 =
nX
j=1
jfj ajj2 + 2
mX
j=1
jgj bjj2
L0 =
nX
j=1
jfj an j+1j2 + 2
mX
j=1
jgj +bjj2:
Proof. Note that the subspaces of symmetric matrices and skew symmetric matrices of Rn n
are orthogonal with respect to the inner product (X;Y) = trXY>. So
kkAk 1 +hBh 1 Ck2 =kkAk 1 (C +C>)=2k2 +khBh 1 (C C>)=2k2:
Hence
?0 = min
k2SO(n)
kkAk 1 (C +C>)=2k2 + min
h2SO(n)
khBh 1 (C C>)=2k2:
37
Consider n = 2m. For the rst term using (3.28) (dropping the scalar 2n and with B = 0
and C C> = 0 in mind) and for the second term using [34, Theorem 2.1] (with A = 0
and C + C> = 0 in mind), we have ?0. Similarly from (3.29) we have L0. The odd case is
simpler.
Remark 3.14. Theorem 3.11 is also true for reductive Lie groups with connected K (so
the Weyl group Wt is de ned). For example, the reductive group (GLn(C);U(n); ;B) with
(X) = X and B(X;Y) = trXY yields [34, Theorem 2.1].
The reductive group (GLn(R);O(n); ;B) with (X) = X> and B(X;Y) = trXY
has non-connected K = O(n). Now k = so(n) and p is the space of n n real symmetric
matrices. Though K = O(n) is not connected, O(n) = SO(n)[ ^SO(n) where
^SO(n) = diag ( 1;1;:::;1)SO(n)
is the set of orthogonal matrices with determinant 1. So we cannot apply Theorem 3.11
directly. But the problem can be solved by two approaches. The rst approach is to use
O(n) = SO(n)[ ^SO(n) and apply Example 3.12. The second approach is to use O(n) U(n)
and apply [34, Theorem 2.1] as follows.
We only consider even n since the odd case is similar. Let A2p, B2k and C2gln(R).
Then
C = U
2
64
0
B@ 0 g1
g1 0
1
CA
0
B@ 0 gm
gm 0
1
CA
3
75U 1 +Vdiag (f
1;:::;fn)V 1 (3.34)
for some U;V 2O(n), g1 gm 0, and f1 fn. Moreover
A = Zdiag (a1;:::;an)Z 1 (3.35)
38
and
B = W
2
64
0
B@ 0 b1
b1 0
1
CA
0
B@ 0 bm
bm 0
1
CA
3
75W 1 (3.36)
for some Z;W 2 O(n), b1 bm 0, and a1 an. Since O(n) U(n), by [34,
Theorem 2.1]
min
k;h2O(n)
kkAk 1 +hBh 1 Ck2 min
k;h2U(n)
kkAk 1 +hBh 1 Ck2
=
nX
j=1
jfj ajj2 + 2
mX
j=1
jgj bjj2;
where k k denotes the Frobenius norm. Clearly the right side is attainable so we have
min
k;h2O(n)
kkAk 1 +hBh 1 Ck2 =
nX
j=1
jfj ajj2 + 2
mX
j=1
jgj bjj2: (3.37)
Similarly
max
k;h2O(n)
kkAk 1 +hBh 1 Ck2 =
nX
j=1
jfj an j+1j2 + 2
mX
j=1
jgj +bjj2: (3.38)
The outcomes are identical to those [34, Theorem 2.1] if we view A;B;C2gln(C).
The following summaries the above discussion and asserts that the set fkkAk 1 +
hBh 1 Ck2 : k;h2O(n)g is connected though O(n) is not.
Theorem 3.15. Let C 2 Rn n, A 2 Rn n be symmetric with eigenvalues a1
an and B 2 Rn n be skew symmetric with eigenvalues ib1;:::; ibm when n = 2m and
ib1;:::; ibm;0 when n = 2m + 1 (b1 bm 0). Let A;B;C have decompositions
(3.35), (3.36) and (3.34) respectively. Let k k be the Frobenius norm on Rn n. Then
fkkAk 1 +hBh 1 Ck2 : k;h2O(n)g= [?;L];
39
where
? =
nX
j=1
jfj ajj2 + 2
mX
j=1
jgj bjj2
L =
nX
j=1
jfj an j+1j2 + 2
mX
j=1
jgj +bjj2:
Proof. We only deal with the even case n = 2m (the odd case is simpler). We have already
established (3.37) and (3.38). Let
S :=fkkAk 1 +hBh 1 Ck2 : k;h2O(n)g:
The set S contains
S1 =fkkA1k 1 +hB1h 1 C1k2 : k;h2SO(n)g
in which A1;B1;C1 are A;B;C as in (3.35) (3.36), (3.34) respectively and in addition
U;Z;W 2SO(n) (we can also assumeV 2SO(n) otherwise replaceV byVdiag ( 1;1;:::;1)).
Now S1 is an interval since SO(n) is compact connected, i.e., S1 = [?1;L1]. By Theorem 3.13
?1 =
nX
j=1
jfj ajj2 + 2
mX
j=1
jgj bjj2 = ?
and
L1 =
8
>><
>>:
Pn
j=1jfj an j+1j
2 + 2Pm
j=1jgj +bjj
2 if m is even
Pn
j=1jfj an j+1j
2 + 2(Pm 1
j=1 jgj +bjj
2 +jgm bmj2) if m is odd
The set S also contains
S2 =fkkA2k 1 +hB2h 1 C2k2 : k;h2SO(n)g= [?2;L2]
40
in which A2;B2;C2 are A;B;C as in (3.35) (3.36), (3.34) respectively except the last block
B2 is the negative of the last block of B and in addition U;V;Z;W 2SO(n) as before. Again
by Theorem 3.13
?2 =
nX
j=1
jfj ajj2 + 2(
m 1X
j=1
jgj bjj2 +jgm +bmj)
and
L2 =
8>
><
>>:
Pn
j=1jfj an j+1j
2 + 2Pm
j=1jgj +bjj
2 if m is odd
Pn
j=1jfj an j+1j
2 + 2(Pm 1
j=1 jgj +bjj
2 +jgm bmj2) if m is even
Note that L1 ?2 ?1 so S1 and S2 intersect. Moreover L = L1 2S1 if m is even and
L = L2 2S2 if m is odd. We then have the desired result.
3.6 Global Extrema
The gradient ow we obtained in Section 3.3 could be used to design an algorithm to
solve Problem (3.5) as the following coupled discretized gradient system alike [34]:
km+1 = km expf m[ (C Ad (hm)B);Ad (km)A]kg; m = 1;2;:::
hm+1 = hm expf m[ (C Ad (km)A);Ad (hm)B]kg; m = 1;2;:::
where m; m > 0 are the steps. But the gradient ow method has pitfalls of local minima.
We now consider a special case: B = 0 and A;C 2 p. So the problem is to study
the distance between C and the adjoint orbit Ad (K)A of A. Such problem would have a
unique local minimum except for a measure zero set of A and C. Since B = 0 and A;C2p,
Problem (3.5) becomes
min
k2K
kAd (k)A Ck; (3.39)
41
which is equivalent to the problem
max
k2K
B(Ad (k)A;C): (3.40)
Let a be a maximal Abelian subspace of p. Since p = Ad (K)a and the Killing form B is
AdK-invariant, we may assume that A;C2a. De ne a smooth function fC;A : K!R as
fC;A(k) = B(Ad (k)A;C):
This induces a function ^fC;A : K=KA ! R since K=KA Ad (K)A [49, p.214], where
KA :=fk2K : Ad (k)A = Ag is the centralizer of A in K.
The set of non-regular elements in a is of measure zero, since it is the union of nite
many hyperplanes in a. If C is regular, then ^fC;A is a Morse function on K=KA. If in addition
A and C are in the same Weyl chamber, then fC;A has a unique local minimum and local
maximum (which are global minimum and maximum) [49, Proposition 4.4].
As an example, consider G = GLn(C). Without loss of generality we may assume that
A = diag (a1;:::;an) and C = diag (c1;:::;cn) with a1 an and c1 cn. Then
(3.40) becomes
max
U2U(n)
trCUAU : (3.41)
It is well known that for (3.41) the global maximum is Pni=1 aici. The permutation matrix
groupfP : 2Sng U(n) is part of the critical set of fC;A. For each 2Sn, Pni=1 aic i is
a critical value of fC;A. Under the assumption that A and C are regular, i.e., a1 > >an
and c1 > >cn, the optimization problem (3.39) has a unique local (global) minimum [12,
Theorem 4.1]. A similar result is true for GLn(R).
42
Chapter 4
Convexity of Generalized Numerical Ranges Associated with Lie Algebras
This chapter is essentially a brief survey on some generalized numerical ranges associated
with Lie algebras. The classical numerical range of a complex square matrix is the image of
the unit sphere under the quadratic form. One of the most beautiful properties is that the
numerical range of a matrix is always convex. We give another proof of the convexity of a
generalized numerical range associated with a compact Lie group via a connectedness result
of Atiyah and a Hessian index result of Duistermaat, Kolk and Varadarajan.
4.1 Classical Numerical Range
Let Cn (resp., Rn) be the vector space of all n-tuple complex (resp., real) numbers.
Let Cn n denote the set of all complex n n matrices. The (classical) numerical range of
A2Cn n is the set
W(A) =fx Ax : x2Cn;x x = 1g C;
which is the image of the unit sphere in Cn under the quadratic map x7!x Ax. Toeplitz-
Hausdor theorem [48, 22] asserts that W(A) is convex for all A 2 Cn n, which perhaps
is the most interesting property of numerical range. See [13] for an interesting geometric
proof. The following is a collection of some other nice properties of numerical range, for
which proofs and references can be found in [18, 26].
Proposition 4.1. The following statements hold for all A2Cn n.
(1) W(A) is compact.
(2) (A) W(A), where (A) is the spectrum of A.
43
(3) W(A+ I) = W(A) + for all 2C, where I2Cn n is the identity matrix.
(4) W( A) = W(A) for all 2C.
(5) W(U AU) = W(A) for all U2U(n), where U(n) is the unitary group.
(6) W(A) = conv (A) if A is normal, where conv (A) is the convex hull of (A).
(7) W(A B) = conv (W(A)[W(B)) for any B2Ck k with k2N.
(8) W(S) W(A) for any principal submatrix S of A.
(9) If A2C2 2 has eigenvalues 1 and 2, then W(A) is an elliptical disk with 1 and 2
as foci, and minor axis of length ptr (A A) j 1j2 j 2j2.
4.2 Generalized Numerical Ranges
There are many generalizations of the classical numerical range motivated by theo-
ries and applications in the last decades [20, 33]. Halmos [21] introduced the notion of
k-numerical range of A2Cn n for 1 k n, which is de ned by
Wk(A) =
( kX
i=1
x iAxi : x1;:::;xk are orthonormal in Cn
)
:
He conjectured and Berger [2] proved that Wk(A) is always convex. Westwick [51] further
generalized the k-numerical range to the c-numerical range of A2Cn n for c2Cn de ned
by
Wc(A) =
( nX
i=1
cix iAxi : x1;:::;xn are orthonormal in Cn
)
:
Westwick proved that Wc(A) is always convex if c2 Rn and fails to be convex if c2 Cn
in general. Let c = (c1;:::;cn)> and C = diag (c1;:::;cn). Then one sees that 2Wc(A)
if and only if 2 tr (CU AU) for some U 2 U(n), where U(n) denotes the unitary group.
This observation motivates the de nition of C-numerical range of A2Cn n for a general
44
C2Cn n de ned by
WC(A) =ftr (CU AU) : U2U(n)g:
This notion was rst introduced by Goldberg and Straus in [19]. Note that WC(A) is the
image of the unitary orbit
U(A) =fU AU : U2U(n)g:
under the linear functional on Cn n represented by C. Clearly WC(A) = WA(C) and
WC(A) = WC(U AU) for all U 2 U(n). Cheung and Tsing [11] proved that WC(A) is
star-shaped.
4.3 Generalized Numerical Ranges Associated with Lie Algebras
4.3.1 Compact Case
Let C 2 Cn n be Hermitian and let A 2 Cn n. Let A = A1 + iA2 be a Hermitian
decomposition, where A1 and A2 are Hermitian. Then WC(A) can be identi ed with
WC(A1;A2) :=f(trCU A1U;trCU A2U) : U2U(n)g R2:
Note that U(n) is a compact connected Lie group, whose Lie algebra u(n) consists of all
n n skew Hermitian matrices. If B2Cn n is Hermitian, then iB;iC2u(n) and
tr (CU BU) = tr (BUCU ) = tr (iB)U(iC)U :
Thus one can assume that A1;A2;C2u(n) when concerning convexity of WC(A1;A2).
Westwick?s proof uses the idea of Hausdor ?s connectedness argument. He considered
the function fB : U(n)=D(n) ! R given by fB([U]) = trCU BU, where B;C 2 Cn n
are Hermitian, D(n) is the subgroup of diagonal matrices in U(n), and [U] = D(n)U for
45
U 2 U(n). He showed that f 1B (c) is connected for any c2R. Ra s [40] pointed out that
there is a gap in Westwick?s proof since the eigenvalues of C and B are assumed distinct.
Motivated by Westwick?s paper, Ra s [40] considered a generalized numerical range
associated with a compact Lie group. The following is Ra s [40] consideration. Let K be
a compact connected Lie group with Lie algebra k. Let h ; i be any AdK-invariant inner
product on k, i.e.,
hAd (k)X;Ad (k)Yi=hX;Yi; 8X;Y 2k;8k2K:
For any X1;X2;C2k, the C-numerical range of the pair (X1;X2) is de ned by
WC(X1;X2) =f(hX1;Ad (k)Ci;hX2;Ad (k)Ci) : k2Kg:
Tam [46] proved that WC(X1;X2) is convex in R2. One may also consider the joint C-
numerical range of X1;:::;Xp2k de ned by
WC(X1;:::;Xp) =f(hX1;Ad (k)Ci;:::;hXp;Ad (k)Ci) : k2Kg:
Tam?s result is best possible in the sense that WC(X1;:::;Xp) fails to be convex in general
if p 3 [6]. The main ideas in Tam?s proof are applying a connectedness result of Atiyah [1]
and using the symplectic structure of the co-adjoint orbit. Then the connectedness of the
bres of the map C : Ad (K)X!R de ned by
C(Y) =hC;Yi; 8Y 2Ad (K)X
is established. The convexity of WC(X1;X2) then follows through rotation.
Very recently Markus and Tam [37] gave another proof of the convexity of WC(X1;X2).
Without using symplectic technique, they proved the connectedness of the bres of the map
46
fC;X : K!R for all C;X2k de ned by
fC;X(k) =hC;Ad (k)Xi; 8k2K:
The bre connectedness result in the compact group K of Markus and Tam is clearly stronger
than the bre connectedness result in the adjoint orbit Ad (K)X:
K Ad ( )X //
fC;X
Ad (K)X
Czz
R
since the map Ad ( )X : K7!Ad (K)X is continuous.
We shall give a third convexity proof (see Remark 4.10) via a connectedness result of
Atiyah [1] and a Hessian index result of Duistermaat, Kolk and Varadarajan [16].
It is worthwhile to note that one may further assume that K is semisimple when con-
cerning the convexity of WC(X1;X2). Since K is compact, we have K = Z0Ks = KsZ0 [29,
Prop 4.29] and k = ks z, where Ks is semisimple with Lie algebra ks, Z0 is the identity
component of the center Z of K, and z0 is the Lie algebra of Z. Since AdZ is trivial and
AdK acts trivially on z, for any A = As +A0 with As2ks and A0 2z and for any k = ksz0
with ks2Ks and z0 2Z0, we have Ad (k)A = Ad (ks)As +A0. Thus
WC(X1;X2) =f(hX1s;Ad (ks)Csi;hX2s;Ad (ks)Csi) : ks2Ksg+c;
where c2R2 is a constant and X1s;X2s;Cs2ks.
4.3.2 Complex Semisimple Case
Tam [45] considered a generalized C-numerical range in the context of complex semisim-
ple Lie algebras. Let g be a complex semisimple Lie algebra and let k be a compact real form
of g. Then g = k ik is a Cartan decomposition of g with a corresponding Cartan involution
47
. The Killing form B induces an inner product B on g de ned by
B (X;Y) = B(X; Y); 8X;Y 2g:
Let G be a connected complex Lie group with Lie algebra g and let K be the analytic
subgroup of G with Lie algebra k. Given X;C 2g, the C-numerical range of X is de ned
by
WC(X) =fB (C;Ad (k)X) : k2Kg: (4.1)
Note that the usual C-numerical range is for the reductive Lie algebra gln(C) and that the
compact case is essentially a special one with X 2 k. Tam [45] conjectured that for any
X 2g and f 2g , the dual space of g, the set f(Ad (K)X) is star-shaped with respect to
the origin.
The adjoint orbit Ad (K)X depends only on AdGK, the analytic subgroup of the adjoint
group Int g corresponding to ad gk, and thus Ad (K)X is independent of the choice of G. Let
t be a maximal abelian subalgebra of k. Then h = t it is a Cartan subalgebra of g. Let
g = h
M
2
g be the root space decomposition of g with respect to h, where
g =fX2g : [H;X] = (H)X for all H2hg;
=f 2h : 6= 0 and dim g 6= 0g:
Since B(g ;g ) = 0 whenever + 6= 0, we have the orthogonal projection : g ! h.
Cheung and Tam [10] proved that (Ad (K)X) is star-shaped in h with star center 0 for all
X2g. They further a rmed Tam?s conjecture for the complex simple Lie algebras of type
B [10]. The conjecture is valid for simple Lie algebras of type A [11], D;E6 and E7 [15]; it
remains unknown for type C;E8;F4 and G2.
48
4.3.3 Real Semisimple Case
Li and Tam [35] generalized C-numerical range in the context of real semisimple Lie
algebras. Let g be a real semisimple Lie algebra. Let G be a connected real semisimple Lie
group with Lie algebra g. Let g = k p be the Cartan decomposition of g corresponding
to a Cartan involution , where k and p are the +1 and 1 eigenspaces of , respectively.
The Killing form B is positive de nite on p and negative de nite on k. Let K be the
analytic subgroup of G with Lie algebra k. For C;X1;:::;Xp2p, the C-numerical range of
(X1;:::;Xp) is de ned by
WC(X1;:::;Xp) =f(B(C;Ad (k)X1);:::;B(C;Ad (k)Xp)) : k2Kg: (4.2)
Since Ad (K)X is independent of the choice of connected G, so is (4.2). Li and Tam [35]
proved thatWC(X1;X2) is convex for all classical real simple Lie algebras except sl2(R). They
also investigated WC(X1;X2;X3) case by case for each classical real simple Lie algebra. It
would be nice if we can show the convexity results of [35] in a uni ed way.
Remark 4.2. Reductive Lie algebras have similar structures with semisimple ones (see
Remark 3.3). Thus the C-numerical range (4.2) is also well de ned for reductive Lie algebras.
We begin with the notation of Morse function [38]. Let M be a manifold and f : M!R
a smooth function. A point p 2 M is called a critical point of f if the di erential map
dfp : Tp(M)!Tf(p)(R) is trivial. If p is a critical point of f, the Hessian Hp of f at p is a
symmetric bilinear form on Tp(M) de ned by
Hp(v;w) = Vp(Wf); 8v;w2Tp(M);
where V and W are vector elds extended by v and w (i.e., Vp = v;Wp = w), respectively,
and where Wf 2C1(M) is de ned by (Wf)(q) = Wq(f) for all q 2M. It is symmetric
49
because
Vp(Wf) Wp(Vf) = [V;W]p(f) = 0
where [V;W] is the Poisson bracket of V and W, and where [V;W]p(f) = 0 since p is a
critical point. It is well-de ned because Vp(Wf) = v(Wf) is independent of the extension
V of v, while Wp(Vf) is independent of W. If we choose a local chart about p, the Hessian
can be represented by a real symmetric matrix. The index of Hp, referred to as the index of
f at p, is the maximal dimension of a subspace of Tp(M) on which Hp is negative de nite, or
equivalently the number of negative eigenvalues of the matrix associated with Hp. A smooth
function on a manifold is called a Morse function if its Hessian is nondegenerate at every
critical point. A Morse-Bott function [4, 5] is a smooth function on a manifold whose critical
set is a closed submanifold and whose Hessian is nondegenerate in the normal direction.
Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the
critical submanifold.
Now let a be a maximal Abelian subspace of p and let g = k a n be a Iwasawa
decomposition of g. Let G = KAN be the corresponding Iwasawa decomposition of G. Let
W = W(G;A) = M0=M be the Weyl group of G relative to A, where M0 and M are the
normalizer and centralizer of A in K, respectively. For C;X 2 p, we consider the smooth
function fC;X : K!R de ned by
fC;X(k) = B(C;Ad (k)X): (4.3)
The C-numerical range WC(X;Y) with C;X;Y 2 p is convex if every bre f 1C;X(c) with
c2R is connected (or empty) in K. Since p = [k2KAd (k)a and B is AdK-invariant, we
can assume that C;X2a.
Example 4.3. Let g = sl2(R) and G = SL2(R). Up to a multiple of 4, the Killing form
is given by B(X;Y) = trXY for all X;Y 2 g. Let the Cartan involution be de ned by
50
(X) = X> for all X2g. Then
p =
8
><
>:
0
B@a b
b a
1
CA : a;b2R
9
>=
>;
k = so(2) =
8
><
>:
0
B@ 0 c
c 0
1
CA : c2R
9
>=
>;
K = SO(2) =
8
><
>:
0
B@ cos sin
sin cos
1
CA : 2R
9
>=
>;:
Let a =
8>
<
>:
0
B@a 0
0 a
1
CA : a2R
9>
=
>;. Pick C = X =
0
B@1 0
0 1
1
CA 2 a; Y =
0
B@0 1
1 0
1
CA 2 p.
For k =
0
B@ cos sin
sin cos
1
CA, we have f
C;X(k) = tr (CkXk 1) = 2 cos 2 . Thus the bre
f 1C;X(2) =
8
><
>:
0
B@1 0
0 1
1
CA;
0
B@ 1 0
0 1
1
CA
9
>=
>; is clearly not connected in K. In fact, the C-numerical
range of (X;Y) is not convex. More precisely, WC(X;Y) =f(2 cos 2 ;2 sin 2 ) : 2Rg is a
circle on R2.
For eachX2a, letKX andWX denote the centralizers ofX inK and inW, respectively.
It is obviously that M KX, which guarantees that the notion KCwKX makes sense for
w2W. The following two lemmas show that fC;X is a Morse-Bott function.
Lemma 4.4. ([16, p.314{316], [49, p.214]) The critical set of fC;X is
KC;X =fk2K : [C;Ad (k)X] = 0g
=
[
w2W
KCwKX
=
[
w2WCnW=WX
KCwKX
51
where the second union is disjoint and over a complete set of double coset representatives.
Lemma 4.5. ([16, p.317] [49, p.216]) Let k = uxwv with u 2 KC, v 2 KX, and xw a
representative of w in K. The Hessian Hk of fC;X at k2K is given by
Hk(dLk(Z);dLk(Z))
= d
2
dt2
t=0
fC;X(kexptZ)
=
X
2 +
(X)(w )(C)kF (Ad (v)Z)k2; 8Z2k (4.4)
where dLk : k !Tk(K) denotes the di erential at the identity of the left translation Lk :
K !K given by Lk(h) = kh and F : k ! k is an orthogonal projection. In particular,
fC;X is a Morse-Bott function and its index at k is
X
2 +; (X)(w )(C)>0
dim g : (4.5)
Remark 4.6. For each 2 +, de ne kv = fZ 2 k : Ad (v)Z 2 k g. On each subspace
kv of k, depending on the value of (X)(w )(C), exactly one of the following three cases
happens for the Hessian: (1) positive de nite, (2) negative de nite, (3) trivial.
Noting that Adv : k!k is nonsingular and that k = m
M
2 +
k (see Lemma 2.4), we
see dim kv = dim k = dim g and k = Ad (v 1)m
M
2 +
kv . The index of fC;X is then
X
2 +; (X)(w )(C)>0
dim kv =
X
2 +; (X)(w )(C)>0
dim g :
The following example shows the explicit expression of the Hessian of fC;X for sln(C).
Example 4.7. Let g = sln(C) be viewed as a real semisimple Lie algebra. The Killing form
of g is given by B(X;Y) = Re trXY for all X;Y 2 g up to a scalar multiple of 4n. Let
the Cartan involution on g be de ned by (X) = X for all X 2 g. Then k = su(n),
K = SU(n), and p consists of Hermitian matrices in g. Let a p be the subspace of (real)
52
diagonal matrices. The root space decomposition of g with respect to a is
g = (a ia)
M
i6=j
CEij;
where Eij is the matrix with 1 at the (i;j)-entry and 0 elsewhere. The root system is
= fei ej : 1 i6= j ng, where ei 2 a sends A2 a to the i-th diagonal entry of
A. The Weyl group W is isomorphic to the group Pn of permutation matrices. Let a+ a
be the fundamental Weyl chamber consisting of all diagonal matrices whose diagonal entries
are in descending order. The set of positive roots is then + = fei ej : 1 i < j ng.
For each = ei ej 2 +, k = fcEij cEji : c2 Cg. Pick C = diag (c1;:::;cn);X =
diag (x1;:::;xn) 2 a. The centralizers KC (resp., KX) of C (resp., X) in K consists of
all matrices in SU(n) that commute with C (resp., X). The critical set of fC;X is thus
KC;X = KCPnKX. For each k = UPV with U2KC, P 2Pn, and V 2KX, the Hessian of
fC;X at k is
X
2 +
(w )(C) (X)kF (Ad (v)Z)k2; Z2su(n)
= 8n
X
i0
dimR(CEij)
= 2 jf(i;j) : 1 i 0gj
The following lemma of Atiyah is crucial.
53
Lemma 4.8. [1, p.4] Let f : M ! R be a Morse-Bott function on a compact connected
manifold M. If neither f nor f has a critical manifold of index 1, then f 1(c) is connected
(or empty) for every c2R.
The above lemmas enable one to focus on the computation of the index
X
2 +; (X)(w )(C)>0
dim g
of fC;X in (4.5) and the index
X
2 +; (X)(w )(C)<0
dim g
of fC;X. If neither of them is 1, the convexity of WC(X;Y) follows for real semisimple Lie
groups G with nite center (in which case K is compact). As an application, we have the
convexity of the C-numerical range for complex semisimple Lie groups.
Theorem 4.9. Let G be a complex semisimple Lie group viewed as a real Lie group, and
let g = k p with p = ik be a Cartan decomposition of the (real) Lie algebra g of G. Then
the C-numerical range WC(X;Y) de ned in (4.2) is convex for all C;X;Y 2p.
Proof. Note that K is compact since k is a compact real form of g. Since g is complex
semisimple, each g has even dimension over R [49, p.217] and thus the indices of fC;X and
fC;X are both even. By Atiyah?s lemma f 1C;X(c) is connected for all c 2 R. Rotating
WC(X;Y) anti-clockwise by an angle 2R yields WC(X0;Y0), where
(X0;Y0) = (cos X + sin Y; sin X + cos Y)2p p:
It follows that the intersection of WC(X;Y) with every straight line is connected, whence
WC(X;Y) is convex.
54
Remark 4.10. Since p = ik in Theorem 4.9. It is essentially the same as the compact case
discussed in Section 4.3.1. This gives a third proof of Tam?s result in [46].
The following example shows that the index condition is su cient but not necessary for
convexity of C-numerical range.
Example 4.11. Let g = sl3(R) and G = SL3(R). Let the Cartan involution be de ned
by (X) = X> for all X 2 g. Then k = so(3), K = SO(3), and p is the space of all
traceless symmetric matrices. Let a p be the subspace of diagonal matrices. The root
space decomposition of g relative to a is
g = a
M
i6=j
REij:
The root system is = fei ej : 1 i6= j 3g. The centralizer M of a in K consists of
diagonal matrices in SO(3), and the normalizer of a in K consists of generalized permutation
matrices in SO(3) whose nonzero entries are either 1 or 1. Thus the Weyl group W = M0=M
is isomorphic to the group P3 of permutation matrices. Let a+ a be the fundamental Weyl
chamber consisting of all diagonal matrices whose diagonal entries are in descending order.
The set of positive roots is then + = fe1 e2;e2 e3;e1 e3g. Now pick C;X2a+ and
consider the map fC;X : SO(3) ! R as de ned in (4.3). Obviously KC = KX = M. By
Lemma 4.4, the critical manifold of fC;X is KC;X = KCWKX = M0. Because (X) > 0 and
dim g = 1 for all 2 +, the index given by (4.5) is equal to the number of positive roots
sent to positive roots by the w2W under consideration. Since WC and WX are trivial, each
Weyl group element can appear for some k 2 KC;X. Therefore we have the following six
cases for the index of fC;X with the notations that 1 := e1 e2; 2 := e2 e3; 3 := e1 e3.
Case 1: w =
0
BB
BB
@
1 0 0
0 1 0
0 0 1
1
CC
CC
A
. Since w 1 = 1, w 2 = 2, w 3 = 3, the index is 3.
55
Case 2: w =
0
BB
BB
@
0 1 0
1 0 0
0 0 1
1
CC
CC
A
. Since w 1 = 1, w 2 = 3, w 3 = 2, the index is 2.
Case 3: w =
0
BB
BB
@
1 0 0
0 0 1
0 1 0
1
CC
CC
A
. Since w 1 = 3, w 2 = 2, w 3 = 1, the index is 2.
Case 4: w =
0
BB
BB
@
0 0 1
0 1 0
1 0 0
1
CC
CC
A
. Since w 1 = 2, w 2 = 1, w 3 = 3, the index is
0.
Case 5: w =
0
BB
BB
@
0 0 1
1 0 0
0 1 0
1
CC
CC
A
. Since w 1 = 2, w 2 = 3, w 3 = 1, the index is 1.
Case 6: w =
0
BB
BB
@
0 1 0
0 0 1
1 0 0
1
CC
CC
A
. Since w 1 = 3, w 2 = 1, w 3 = 2, the index is 1.
Now dimK = dim SO(3) = 3. This shows that for g = sl3(R) the condition of Lemma
4.8 is not satis ed, but WC(X;Y) is still convex [6].
56
Bibliography
[1] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 308
(1982), 1{15.
[2] C. A. Berger, Normal Dilations, Ph.D. Thesis, Cornell University, 1963.
[3] A. Bloch, Ed., Hamiltonian and Gradient Flows, Algorithms and Control, Fields Institute
Communications, Amer. Math. Soc., Providence, RI, 1994.
[4] R. Bott, Nondegenerate critical manifolds, Ann. of Math., 60 (1954), 248{261.
[5] R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. (N. S.), 7 (1982)
331{358.
[6] L. Brickman, On the eld of values of a matrix, Proc. Amer. Math. Soc., 12 (1961),
61{66.
[7] T. Br ocker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag,
New York, 1985.
[8] R. W. Brockett, Dynamical systems that sort lists, diagonalize matrices, and solve linear
programming problems, Linear Algebra Appl., 146 (1991), 79{91.
[9] R. W. Brockett, Di erential geometry and the design of gradient algorithms, Proc. Symp.
Pure Math., 54 (1993), 69{92.
[10] W. S. Cheung and T. Y. Tam, Star-shapedness and K-orbits in complex semisimple Lie
algebras, Canad. Math. Bull., 54 (2011), 44{55.
[11] W. S. Cheung and N. K. Tsing, The C-numerical range of matrices is star-shaped,
Linear Multilinear Algebra, 41 (1996), 245{250.
[12] M. T. Chu and K. Driessel, The projected gradient method for least squares matrix
approximations with spectral constraints, SIAM J. Numer. Anal., 27 (1990), 1050{1060.
[13] C. Davis, The Toeplitz-Hausdor theorem explained, Canad. Math. Bull., 14 (1971),
245{246.
[14] J. Day, W. So and R. C. Thompson, The spectrum of a Hermitian matrix sum, Linear
Algebra Appl., 280 (1998), 289{332.
[15] D.Z. Djokovic and T.Y. Tam, Some questions about semisimple Lie groups originating
in matrix theory, Canad. Math. Bull., 46 (2003), 332{343.
57
[16] J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan, Functions, ows and oscil-
latory integrals on ag manifolds and conjugacy classes in real semisimple Lie groups,
Compositio Math., 49 (1983), 309{398.
[17] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull.
Amer. Math. Soc., 37 (2000), 209{249.
[18] M. Goldberg, On certain nite-dimensional numerical ranges and numerical radii, Linear
Multilinear Algebra, 7 (1979), 329{342.
[19] M. Goldberg and E. G. Straus, Elementary inclusion relations for generalized numerical
ranges, Linear Algebra Appl., 18 (1977), 1{24.
[20] K. E. Gustafson and D. K. M. Rao, Numerical Range: The Field of Values of Linear
Operators and Matrices, Springer, New York, 1997.
[21] P. R. Halmos, A Hilbert Space Problem Book, 2ed., Springer-Verlag, New York, 1982.
[22] F. Hausdor , Der Wertvorrat einer Bilinearform, Math Z., 3 (1919), 314{316.
[23] S. Helgason, Di erential Geometry, Lie Groups, and Symmetric Spaces, Academic
Press, New York, 1978.
[24] U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer-Verlag,
London, 1994.
[25] R. R. Holmes and T. Y. Tam, Distance to the convex hull of an orbit under the action
of a compact Lie group, J. Austral. Math. Soc., (Series A) 66 (1999), 331{357.
[26] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press,
Cambridge, 1991.
[27] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-
Verlag, New York, 1972.
[28] J. E. Humphreys, Re ection groups and Coxeter groups, Cambridge University Press,
New York, 1990.
[29] A. W. Knapp, Lie Groups Beyond an Introduction, 2ed., Birkh auser, Boston, 2002.
[30] B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci.
Ecole Norm. Sup., (4) 6 (1973), 413{455.
[31] A. Knutson and T. Tao, Honeycombs and sums of Hermitian matrices, Notices Amer.
Math. Soc., 48 (2001), 175{186.
[32] J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2002.
[33] C. K. Li, C-numerical ranges and C-numerical radii, Linear Multilinear Algebra, 37
(1994), 51{82.
58
[34] C. K. Li, Y. T. Poon, and T. Schulte-Herbr uggen, Least-squares approximation by
elements from matrix orbits achieved by gradient ows on compact Lie groups, Math.
Comp., 80 (2011), 1601{1621.
[35] C. K. Li and T. Y. Tam, Numerical ranges arising from simple Lie algebras, Canad. J.
Math., 52 (2000), 141{171.
[36] M. Liao, L evy Processes in Lie Groups, Cambridge University Press, 2004.
[37] A. Markus and T. Y. Tam, Connectedness of some bres on a compact connected Lie
group, Linear Multilinear Algebra, 59 (2011), 1121{1126.
[38] J. Milnor, Morse theory, Princeton University Press, N.J. 1963.
[39] A. L. Onishchik and E. B. Vinberg, Lie Groups and Lie Algebras III: Structure of Lie
Groups and Lie Algebras, Springer-Verlag, Berlin, 1994.
[40] M. Ra s, Remarques sur un theoreme de R. Westwick, unpublished manuscript.
[41] L. O?Shea and R. Sjamaar, Moment maps and Riemannian symmetric pairs, Math.
Ann., 317 (2000), 415{457.
[42] H. Samelson, Notes on Lie Algebras, Springer-Verlag, New York, 1990.
[43] T. Schulte-Herbruggen, S. J. Glaser, G. Dirr, and U. Helmke, Gradient ows for opti-
mization in quantum information and quantum dynamics: foundations and applications,
Rev. Math. Phys., 22 (2010), 597{667.
[44] T. Y. Tam, A Lie-theoretic approach to Thompson?s theorems on singular values-
diagonal elements and some related results, J. London Math. Soc., (2) 60 (1999), 431{448.
[45] T. Y. Tam, On the shape of numerical ranges associated with Lie groups, Taiwanese J.
Math., 5 (2001), 497{506.
[46] T. Y. Tam, Convexity of generalized numerical range associated with a compact Lie
group, J. Austral. Math. Soc., 72 (2002), 57{66.
[47] T. Y. Tam, Gradient ows and double bracket equations, Di erential Geom. Appl., 20
(2004), 209{224.
[48] O. Toeplitz, Das algebraishe Analogon zu einem Satze von Fej er, Math. Z., 2 (1918),
187{197.
[49] V. S. Varadarajan, The Method of Stationary Phase and Applications to Geometry and
Analysis on Lie Groups, Book Chapter in Algebraic and Analytic Methods in Represen-
tation Theory, Academic Press, 1997, 167{242.
[50] F. Warner, Foundations of Di erentiable Manifolds and Lie Groups, Springer-Verlag,
New York, 1983.
59
[51] R. Westwick, A theorem on numerical range, Linear and Multilinear Algebra, 2 (1975),
311{315.
[52] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Di er-
entialgleichungen, Math. Ann., 71 (1912), 441{479.
[53] A. Wright, Sums of adjoint orbits and L2-singular dichotomy for SU(m), Adv. Math.,
227 (2011), 253{266.
60