Asymptotic Results in Noncompact Semisimple Lie Groups
by
Mary Clair Thompson
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
May 5, 2013
Keywords: Lie group, Lie algebra, semisimple, Aluthge iteration, Bruhat iteration
Copyright 2013 by Mary Clair Thompson
Approved by
Tin-Yau Tam, Chair, Professor of Mathematics
Stewart Baldwin, Professor of Mathematics
Randall R. Holmes, Professor of Mathematics
Huajun Huang, Associate Professor of Mathematics
Abstract
This dissertation is a collection of results on analysis on real noncompact semisimple
Lie groups. Speci?cally, we examine the convergence patterns of sequences arising from the
special group decompositions that exist in this setting.
The dissertation consists of ?ve chapters, the ?rst of which provides a brief introduction
to the topics to be studied.
In Chapter 2, we introduce preliminary ideas and de?nitions regarding Lie groups, their
Lie algebras, and the relationships between the two structures.
Chapter 3 speci?cally examines semisimple Lie algebras and groups; we discuss local
(algebra level) and global (group level) decompositions of real and complex semisimple Lie
groups, such as the root space decomposition of a complex Lie algebra; the local and global
Cartan and Iwasawa decompositions over R; the global Bruhat decomposition; and the
restricted root space decomposition of a real Lie algebra. Each of these will play important
roles in the remainder.
Chapter 4 presents the iterated Aluthge sequence on Cn?n, and extends the sequence to
a real noncompact semisimple Lie group. We use the Cartan decomposition and properties
of the group and its adjoint map to show that the iterated Aluthge sequence converges in
this setting.
The ?nal chapter discusses the matrix iterated Aluthge sequence and its Lie group
generalization using the Bruhat decomposition. We establish convergence of the sequence
(under some conditions) in this general setting using the many special properties of the
decomposition.
ii
Acknowledgments
My most important acknowledgement is to my family, who have never questioned my
desire to pursue mathematics, but have instead constantly supported my decisions. They
have listened to my complaints, fears, and woes with unfaltering patience, and have believed
in my mathematical abilities even when I did not. Their love has been a sustaining force for
me throughout my time as a student, and for that I am truly grateful.
The teaching and mentoring that I have received as a graduate student have been
exceptional. In particular, I wish to thank my advisor, Dr. Tin-Yau Tam, who is a constant
encouragement to me. He is consistently patient with my never-ending list of questions and
never complains when I misunderstand a concept. He has given me the opportunity to assist
him in the research and writing process of multiple papers, and has made it possible for me
to attend conferences and present my research. I could not possibly have chosen a better
advisor.
I would also like to thank the rest of my committee, Dr. Randall Holmes, Dr. Huajun
Huang, and Dr. Stewart Baldwin. I have had the opportunity to learn from each of them
and am certainly a better mathematician because of their teaching.
The atmosphere in the mathematics department at Auburn has been extremely bene?-
cial to me; the professors are open and helpful, fellow graduate students assist me whenever
I ask questions, and the sta? is always willing to help me with problems that arise. The
department has a welcoming aura, with little bickering or competition among graduate stu-
dents or faculty. Without such a supportive department, I would certainly not be in the
?nal stages of preparation of my disseration, and so I am thankful for the many people who
have fostered this environment.
iii
It would be impossible to write an exhaustive list of all of those who have provided
encouragement and support to me throughout the six years I have spent in pursuit of my
Ph.D. Countless friends, students, professors, fellow mathematicians, and even complete
strangers have encouraged me to pursue my love of mathematics and complete my degree. I
would like to thank each one of them for their support.
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Lie Groups and Their Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 8
3 Structure and Decompositions of Semisimple Lie Groups . . . . . . . . . . . . . 16
3.1 Real Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Cartan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Root Space Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Restricted Root Space Decomposition . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Iwasawa Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Weyl Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Bruhat Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.8 Complete Multiplicative Jordan Decomposition . . . . . . . . . . . . . . . . 35
4 Aluthge Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 The ?-Aluthge Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Normal Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Asymptotic Behavior of the Aluthge Sequence . . . . . . . . . . . . . . . . . 43
5 Bruhat Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1 Rutishauser?s LR algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Regular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
v
5.3 Bruhat iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 Open Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5 Comparison of the Iwasawa and Bruhat Iterations . . . . . . . . . . . . . . . 59
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
vi
List of Figures
3.1 Weyl Chambers of a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
vii
Chapter 1
Introduction
The topic of this dissertation is the asymptotic behavior of sequences in noncompact
connected semisimple Lie groups. The motivation for these problems comes from matrix the-
ory: several well-known matrix decompositions lead to matrix sequences whose asymptotic
behavior has been studied extensively. In 2011, Pujals et. al. [7] showed that the iterated
Aluthge sequence, which is based on the polar decomposition of a matrix, converges in Cn?n.
The behavior of the LR sequence, based on the matrix LU decomposition, was studied by
Rutishauser [26]; under certain conditions this sequence also converges. This dissertation
generalizes both of these matrix convergence results to the Lie group setting. Several other
matrix sequences have led to similar generalizations: Holmes et. al. [18] extended a result
on the QR iteration in 2011.
Matrix results provide a natural starting point for the study of general Lie groups.
Well-known matrix groups such as GLn(C) and SLn(C) have a Lie group structure, and the
matrix Lie groups are known as the classical groups; their structure often provides clues to
the structure of more abstract Lie groups. Using the well-studied classical groups as a model
for the abstract groups leads to a much better understanding of the abstract structure than
one might otherwise hope to gain.
The main tools in this dissertation are some important decompositions of semisimple
Lie algebras, which lead to corresponding decompositions on the group level. For example,
the Cartan decomposition of a real semisimple Lie algebra, which is precisely the Hermitian
decomposition on the general linear algebra gln(C), yields the group level Cartan decomposi-
tion (which corresponds to the polar decomposition of the general linear group GLn(C)). In
addition, the Iwasawa decomposition (which corresponds to the matrix QR decomposition)
1
and the Bruhat decomposition (which corresponds to the Gelfand-Naimark decomposition
in matrix theory) are built from factors derived from algebra level decompositions. The
Complete Multiplicative Jordan Decomposition (CMJD) corresponds to the matrix Multi-
plicative Jordan Decomposition. The Cartan, Iwasawa, and Bruhat decompositions exist for
any semisimple Lie group, and indeed for any reductive Lie group.
These decompositions, as well as the tools we use to prove the convergence results, are
available to our use because of the remarkable correspondence between a Lie group and
its Lie algebra. Indeed, the two structures are di?eomorphic via the exponential map in a
neighborhood of the identity, so that results on a Lie algebra often have analogues in the
corresponding Lie groups. For example, the Cartan decomposition of a real semisimple Lie
algebra as g = k?p implies a decomposition of any Lie group G with Lie algebra g: G may
be decomposed asG = KP, whereK is the analytic subgroup generated by k andP = expp.
In particular, properties of the algebra imply that the map K ? expp ? G is actually a
di?eomorphism onto G. Thus the relatively accessible Lie algebra provides a great deal of
insight into the structure of G.
The other group decompositions relevant to this dissertation are all derived from the
algebra-group correspondence, but these decompositions are not the only beautiful relation-
ships between the two structures. As a particularly relevant example, the Weyl group of a
Lie group, which captures the symmetry of the group, and the Weyl group of its Lie alge-
bra, which re?ects the algebra?s symmetry, are actually isomorphic. Again, we see that the
correspondence between the two structures is wonderfully rich.
The organization of this dissertation is as follows: in Chapter 2, we introduce the basic
Lie group and Lie algebra de?nitions and record a few well known results for future reference.
In Chapter 3, we discuss the structure of semisimple Lie groups and the relevant Lie group
decompositions that lead to questions about sequences. Chapter 4 presents the Aluthge
sequence and shows that the sequence converges in a real noncompact semisimple Lie group.
Finally, in Chapter 5, we discuss the Bruhat sequence and the conditions under which it
2
converges in a real noncompact semisimple Lie group. We also consider the problem of
convergence of the sequence under some relaxed conditions and present some illustrative
matrix examples.
3
Chapter 2
Lie Groups and Lie Algebras
In this chapter, we introduce the background information and notation needed through-
out the dissertation. At the most fundamental level, a Lie group is an object that has the
structure of a smooth manifold as well as that of a group; in particular, the two di?erent
structures (one analytic and one algebraic) are tied together by the requirement that the
group operations be smooth mappings on the manifold. Accordingly, we begin by discussing
smooth manifolds, following the treatment in [13], [24], and [34].
2.1 Smooth Manifolds
A topological manifold is a topological space M that has the local structure of n-
dimensional Euclidean space. More precisely, M is a topological manifold of dimension
n if
(1) M is a second countable Hausdor? space, and
(2) every point of M has an open neighborhood that is homeomorphic to an open subset
of Rn.
If M satis?es the properties above, then it is natural to consider the local homeomorphisms,
and in particular the relationships between such homeomorphisms. A chart on M is a pair
(U,?) consisting of an open subset U ? M and a homeomorphism ? : U ? Rn. We should
investigate the behavior of the maps on overlapping charts. To do so, we will need to consider
maps on open subsets of Rn. IfU andV are open in Rn and Rm, respectively, andF is a map
F : U ?V, we say that F is smooth or C? if each component function of F has continuous
partial derivatives of all orders. If, in addition, F is bijective and F?1 is smooth, we say that
4
F is a di?eomorphism. Returning to the manifold M, suppose that charts (U,?) and (V,?)
are chosen so that U ?V ?= ?. Then the transition map ????1 : ?(U ?V) ? ?(U ?V)
is a map between two open subsets of Rn. Two charts (U,?) and (V,?) are called smoothly
compatible if either U ?V = ? or ????1 is C?.
A collection A of charts that cover M is called an atlas, and if each pair of charts in
the atlas A is smoothly compatible, we say that A is a smooth atlas. If A is a maximal
smooth atlas (in the sense that the addition of any chart (U,?) to A would destroy its
property of smooth compatibility) we say that A is a smooth structure on M. Concisely,
A = {(U ,? ) : ??I} is a smooth structure if
(1)
?
?I
U = M,
(2) ? ???1 : Rn ? Rn is C? for all ?,? ?I, and
(3) the collection is maximal with respect to (2).
The topological manifold M is called a smooth manifold (or simply a manifold) if it has
a smooth structure A, and a chart on M is said to be smooth if it is a member of a smooth
structure on M.
We would like to extend the idea of smoothness of maps on Euclidean space to maps on
manifolds. Accordingly, let M be an m-dimensional manifold and N be an n-dimensional
manifold with smooth structures Am and An, respectively, and F : M ? N a continuous
map. The map F is said to be smooth if, for every p?M, there are charts (U,?) ? Am and
(V,?) ? An so that p ? U, F(U) ? V, and ??F ???1 (which maps an open subset of Rm
to an open subset of Rn) is C? from ?(U) to ?(V). In particular, consider the case where
N = R. Then F is called a smooth function on M if for each p?M, there is a corresponding
smooth chart (U,?) so that p?U and F ???1 is C?.
The manifold analogue for the collection of C? functions on open subsets of Euclidean
space is C?(M), which we use to denote the set of all smooth functions F : M ? R. It
is clear that C?(M) is a vector space. It becomes a ring if we introduce the product f ?g,
5
which is calculated by pointwise multiplication. Next, we introduce calculus on M: any
linear map v : C?(M) ? R that satis?es the product rule at some p?M, i.e.,
v(fg) = f(p)v(g) +v(f)g(p), ?f,g ?C?(M),
is called a tangent vector of M at p. The collection of tangent vectors at the point p is a
vector space denoted by Tp(M), known as the tangent space to M at p. Each v ? Tp(M)
(i.e., a vector in the tangent space ofM atp) is called a tangent vector atp. We may think of
M = Rn as a prototype; the tangent vector v acts on f ?C?(Rn) by taking the directional
derivative of f in the direction provided by v, i.e., v(f) = ??f,v?.
Given a point p ? M and a smooth map F : M ? N we assign to F a corresponding
linear map between the tangent spacesTp(M) andTF(p)(N). This is known as the di?erential
(or derivative) of F: the di?erential dFp : Tp(M) ? TF(p)(N) of F at p is the linear map
de?ned by
dFp(v)(f) = v(f ?F), for all v ?Tp(M),f ?C?(N).
Notice that dFp sends a tangent vector v ? Tp(M) to a tangent vector in Tp(N), so that
dFp(v) : C?(N) ? R.
Example 2.1. If M = N = R, then under the natural identi?cation Tp(R) ?= R, the map
dFp : R ? R is just multiplication by F?(p), where F? is the usual derivative of elementary
calculus.
In a sense, the map dFp provides a ?linear approximation? for F or a ?linearization? of
F. The rank of the linear map dFp is well-de?ned, so that we may de?ne the rank of F as
rankdFp. If F is a smooth map so that rankF = dimM for each p?P, then F is called an
immersion. Equivalently, F is an immersion if dFp is injective for each p?M.
We de?ne the concept of a submanifold of M by beginning with a subset S ? M that
is a smooth manifold in its own right; in addition, we want the topological and di?erential
6
structure of S to align with that of M, so we require the inclusion map ? : S ?M to be an
immersion.
At each point p?M, we have de?ned a tangent space Tp(M). The disjoint union
T(M) = ?p?MTp(M)
is called the tangent bundle of M. We de?ne the natural projection ? : T(M) ? M of the
tangent bundle onto its manifold via ?(X) = p for X ? Tp(M). Of course, we may simply
view T(M) as a collection of vector spaces, but the de?nitions actually equip the tangent
bundle with quite a bit of structure: T(M) has a natural topology as well as a smooth
structure derived from M, so that the tangent bundle of a manifold is again a smooth
manifold and in particular, ? : T(M) ?M is smooth.
With tangent spaces and the tangent bundle in place, we may assign to each p ? M a
vector in Tp(M). Such a map X : M ? T(M), so that X(p) ? Tp(M) for all p ? M, is
called a vector ?eld on M. The set of smooth vector ?elds on M is clearly a vector space
over R, and in fact forms a module over the ring C?(M): if X is a vector ?eld on M and
f ?C?(M), then fX is the vector ?eld de?ned by (fX)(p) = f(p)Xp.
A derivation of an algebra A over a ?eld F is a map D : A ? A that satis?es the
following properties:
(1) D(af +bg) = aD(f) +bD(g) for all a,b? F, f,g ? A;
(2) D(fg) = f(Dg) + (Df)g for all f,g ? A.
Any smooth vector ?eld X may be thought of as a derivation of C?(M): let Xf ?
C?(M) be given by Xf(p) = Xpf. Then since Xp ?Tp(M),
X(fg)(p) = Xpfg = f(p)(Xpg) + (Xpf)g(p) = f(p)Xg(p) +Xf(p)g(p).
7
In other words, X(fg) = f(Xg) + (Xf)g. Each derivation of C?(M) can actually be
identi?ed with a smooth vector ?eld: a function X : C?(M) ? C?(M) is a derivation if
and only if it is of the form X(f) = Xf for some smooth vector ?eld X on M [24, p.86]. If
X and Y are smooth vector ?elds on M, the composition X ?Y : C?(M) ? C?(M) may
not be a smooth vector ?eld; however, the Lie bracket [X,Y] := X ?Y ?Y ?X always is.
2.2 Lie Groups and Their Lie Algebras
Let g be a vector space over a ?eld F, and equip g with a product g ? g ? g, denoted
(X,Y) 7? [X,Y]. The following properties arise naturally for many choices of [?,?]:
(1) [?,?] is bilinear.
(2) [X,X] = 0 for all X ? g.
(3) The Jacobi identity [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0 holds for all X,Y,Z ? g.
If g and its product [?,?] satisfy the three properties above, we call [X,Y] the Lie bracket
of X and Y, and call g a Lie algebra over F. We will focus our attention on F = R or C in
our future discussion; the corresponding Lie algebra is called a real or complex Lie algebra.
Example 2.2. A familiar example of a Lie algebra is the general linear algebra gl(V), which
consists of all linear operators on the vector space V over F with the Lie bracket de?ned by
[X,Y] = XY ?YX, X,Y ? gl(V).
Example 2.3. The space of smooth vector ?elds on a manifold M, with
[X,Y] = X ?Y ?Y ?X,
has the structure of a Lie algebra over R.
8
Let g and h be Lie algebras. Both g and h have the underlying structure of a vector
space, so that we have the notion of linear transformation from g to h. Considering the
spaces as Lie algebras, we would like to restrict our attention to linear transformations that
preserve the algebra structure. Accordingly, we de?ne a Lie algebra homomorphism as a
linear transformation ? : g ? h such that
?([X,Y]) = [?(X),?(Y)], for all X,Y ? g.
Given a vector space V over the ?eld F, a Lie algebra homomorphism ? : g ? gl(V) is
called a representation of g. Representations are particularly useful, as ?(g) is an algebra of
matrices which retains inherits the bracket operation from g.
Because of the bilinearity of the bracket and the Jacobi identity, the linear transforma-
tion
ad : g ? gl(g)
de?ned by
adX(Y) = [X,Y], X,Y ? g
is a Lie algebra homomorphism and therefore a representation of g, known as the adjoint
representation of g. Clearly the adjoint representation is of prime importance when we study
Lie algebras. A vector subspace s of g that is a Lie algebra in its own right, i.e., [X,Y] ? s
for all X,Y ? s, is called a subalgebra of g; it is called an ideal if [X,Y] ? s for all X ? g
and Y ? s.
A Lie group G is simultaneously 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.
Example 2.4. The set of all n?n nonsingular complex matrices forms a Lie group, called
the general linear group and denoted by GLn(C). Any closed subgroup of GLn(C) is also a Lie
group, called a closed linear group. Due to their dual topological and algebraic structure, Lie
9
groups have a rich anatomy which provides both analytic and algebraic tools for exploration.
Other classical Lie groups are
1. the real general linear group GLn(R) = GLn(C)?Rn?n,
2. the special linear group SLn(C) = {A? Cn?n : detA = 1},
3. the real special linear group SLn(R) = SLn(C)?Rn?n,
4. the unitary group U(n) = {U ? GLn(R) : U?U = I} ? SLn(C),
5. the orthogonal group O(n) = U(n)?Rn?n,
6. the special orthogonal group SO(n) = {O ? SLn(R) : O?O = I} ? O(n),
7. the complex special orthogonal group SOn(C) = {O ? SLn(C) : O?O = I},
8. the complex symplectic group Spn(C) = {g ? SL2n(C) : g?Jng = Jn}, where Jn =?
?? 0 In
?In 0
?
??.
9. the real symplectic group Spn(R) = Spn(C)?Rn?n,
10. the (compact) symplectic group Sp(n) = Sp(n,C) ? U(2n) =
{
g =
?
?? A B
??B ?A
?
?? : g ?
U(2n)
}
.
The groups GLn(C), SLn(C), SLn(R), SOn(C), SO(n), SU(n), U(n), Spn(C) and Sp(n) are
all connected. The group GLn(R) has two components.
Example 2.5. As a more general example, let V be a ?nite dimensional vector space over
C or R. The group GL(V) of vector space automorphisms of V, with multiplication given
by function composition, is a Lie group.
Let G be a Lie group. Given g ? G, we de?ne the left translation Lg : G ? G by
Lg(h) = gh. Since multiplication is smooth, Lg is a di?eomorphism of G. Because of its
10
manifold structure, we may consider the interaction of a smooth vector ?eld X of G with
Lg: X is left-invariant, or Lg related to itself, if
X ?Lg = dLg ?X, for all g ?G.
When X is viewed as a derivation on C?(G), left-invariance is equivalent to
(Xf)?Lg = X(f ?Lg), for all f ?C?(G), g ?G.
The vector space of left-invariant smooth vector ?elds on G is closed under the bracket
given in Example 2.3, so that it forms a Lie algebra g. In particular, we refer to g as the
Lie algebra of G. The natural identi?cation X 7?Xe of the left-invariant vector ?eld X ? g
with its image in the tangent space of G at the identity is readily seen to be a vector space
isomorphism of g with Te(G). Indeed, we create a bracket operation in Te(G) by setting
[Xe,Ye] := [X,Y]e, so that Te(g) is identi?ed with g and may itself be viewed as a Lie
algebra.
Example 2.6. The Lie algebra of the Lie group GL(V) is gl(V). The Lie algebra of
1. the real general linear group GLn(R) is gln(R) = Rn?n,
2. the special linear group SLn(C) is sln(C) = {X ? Cn?n : trX = 0}.
3. the real special linear group SLn(R) is sln(C)?Rn?n,
4. the unitary group U(n) is u = {X ? Cn?n : X? = ?X},
5. the orthogonal group O(n) is so(n) = {X ? Rn?n : X? = ?X},
6. the special orthogonal group SO(n) is also so(n),
11
7. the complex symplectic group Spn(C) is
spn(C) =
{
?
??X1 X2
X3 ?X?1
?
??: X
i ? Cn?n and X2 = X?2 , X3 = X?3
}
8. the real symplectic group Spn(R) is
spn(R) =
{
?
??X1 X2
X3 ?X?1
?
??: X
i ? Rn?n and X2 = X?2 , X3 = X?3
}
9. and the (compact) symplectic group Sp(n) is sp(n,R) = {X ? gl(2n,R) : X?J =
?JX} [21, p. 59].
A smooth map ? : G ? H between the Lie groups G and H that is also a group ho-
momorphism is called a smooth homomorphism. Since a smooth homomorphism ? preserves
both the manifold and the group structure of G, the di?erential map d?e : g ? h between
the corresponding Lie algebras g and h is actually a Lie algebra homomorphism. We simply
write d? for d?e, and d? is called the derived homomorphism of ?.
Let G be a Lie group with Lie algebra g. A one-parameter subgroup of G is a smooth
homomorphism ? : R ? G, where we view R a Lie group under addition. Due to the
theorem of existence and uniqueness of solutions of linear ordinary di?erential equations,
the map ? 7? d?(0) is a bijection of the set of one-parameter subgroups of G onto g [13,
p.103]. For each X ? g, let ?X be the one-parameter subgroup corresponding to X, and
de?ne the exponential map exp = expg : g ? G by exp(X) = ?X(1). Then exp is smooth,
and ?X(t) = exp(tX), so that the one-parameter subgroups are the maps t 7? exptX for
X ? g. The properties of the exponential map lead to the following important theorem.
Theorem 2.7. [13, p.104] There is a neighborhood N0 of 0 in g and an open neighborhood
Ne of e in G so that exp is an smooth di?eomorphism of N0 onto Ne.
12
In a sense, g and G behave alike near their identities. The exponential map interacts
particularly well with smooth homomorphisms ? : G?H; it is natural in that
??expg = exph ?d?. (2.1)
If G is a closed linear group, its exponential map is just the well-known matrix expo-
nential function [21, p.76], with expA given by
expA =
??
n=0
1
n!A
n.
If the submanifold H of G is also a Lie group in its own right, with multiplication
induced by the multiplication on G, H is called a Lie subgroup of G. If H is also closed
in the topology on G, H is called a closed subgroup. We would like to have a correlation
of some kind between Lie subgroups of G and Lie subalgebras of g. The following theorem
displays an elegant correspondence between the two structures [13, p.112, p.115, p.118]:
Theorem 2.8. LetGbe a Lie group with Lie algebra g. IfH is a Lie subgroup ofG, then the
Lie algebra h of H is a subalgebra of g. Moreover, h = {X ? g : exptX ?H for all t? R}.
Each subalgebra of g is the Lie algebra of exactly one connected Lie subgroup of G.
For g ? G, de?ne a map Ig : G ? G by Ig(x) = gxg?1. Since multiplication in a Lie
group is a smooth operation, Ig is a smooth automorphism; its di?erential, denoted Adg, is
a Lie algebra automorphism of g. Then by (2.1), we have
exp(Adg(X)) = g(expX)g?1, g ?G, X ? g. (2.2)
As a particularly nice example, suppose that G is a closed linear group, i.e., a matrix group.
Then Adg is given by
Adg(X) = gXg?1, g ?G.
13
The exponential map is smooth and invertible in a neighborhood of the identity, so that
if X ? g is close to the identity, g 7? AdgX is smooth as a map from a neighborhood of
e in G to g. In other words, g 7? Adg is a smooth map from a neighborhood of e into
GL(g). In addition, since Ig1g2 = Ig1Ig2, we see that Ad(g1g2) = Ad(g1)Ad(g2). Thus Ad
is smooth on all of G, so that Ad : G ? GL(g) is a smooth homomorphism; in particular,
AdG is a Lie subgroup of GL(g). Ad is actually a representation of G, known as the adjoint
representation. The smoothness of Ad leads us to consider its di?erential, which is just
ad : g ? gl(g) [21, p.80]. Another application of (2.1) shows that
Ad(expX) = exp(adX), X ? g. (2.3)
Note that the exponential map on the left of the equality is exp : g ? G, whereas exp on
the right hand side is exp : gl(g) ? GL(g).
The group Autg of all Lie algebra automorphisms of g is a closed subgroup of GL(g).
Since a closed abstract subgroup of a group G is automatically a Lie subgroup [13, p.115],
Autg is a Lie subgroup of GL(g). Its Lie algebra Derg consists of all derivations of g [13,
p.127]. Due to the Jacobi identity, adX : g ? g is a derivation for all X ? g, so that adg is
a subalgebra of Derg. Indeed, adg generates a connected subgroup Intg of Autg [13, p.127],
known as the adjoint group of g. Elements of adg are called inner derivations, and elements
of Intg are called inner automorphisms. If G is a connected group, then Intg = AdG by
(2.3). The adjoint group Intg is advantageous over AdGsince it is independent of the choice
of the connected Lie group G. The Lie algebra g is said to be compact if G is compact or,
equivalently, the adjoint group Intg is compact.
The symmetric bilinear form B on g de?ned by
B(X,Y) = tr(adXadY), X,Y ? g
14
is called the Killing form. Its interaction with the Lie bracket is associative in the sense that
B([X,Y],Z) = B(X,[Y,Z]), X,Y,Z ? g.
If the Killing form has the property that B(X,Y) = 0 for all Y ? g only when X = 0, we
say that B is nondegenerate on g. If ? is an automorphism of g, then
ad(?X) = ??adX ???1
so that 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, Dk+1g := [Dkg,Dkg];
it is nilpotent if Ckg = 0 for some k, where
C0g = g, and Ck+1g = [Ckg,g].
If the (unique) maximal solvable ideal of g, called the radical of g and denoted by Radg,
is trivial, g is said to be semisimple. As semisimple Lie algebras will be a main object of
interest in this dissertation, let us record for future reference some equivalent conditions to
semisimplicity: g is semisimple if and only if its Killing form is nondegenerate. Additionly,
g is semisimple if and only if it is isomorphic to a direct sum of simple algebras.
The algebra g is reductive if its center z(g) = Radg (or, equivalently, [g,g] is semisimple).
A Lie group is called semisimple (simple, reductive, solvable, nilpotent, abelian) if its Lie
algebra is semisimple (simple, reductive, solvable, nilpotent, abelian).
15
Chapter 3
Structure and Decompositions of Semisimple Lie Groups
In this chapter, we discuss the special structure of semisimple groups which leads to the
group decompositions needed in the sequel.
3.1 Real Forms
For any complex Lie algebra g, restricting the base ?eldCtoRallows g to be viewed as a
real Lie algebra gR, called the reali?cation of g. A subalgebra g0 of gR such that gR = g0?ig0
is called a real form of g, and g is called the complexi?cation of g0. If g0 is a real form of
g, then each Z ? g can be uniquely written as Z = X +iY with X,Y ? g0, and the map
? : g ? g given by
X +iY 7?X ?iY for all X,Y ? g0
is called the conjugation of g with respect to g0. It is clear that
(1) ?2 = 1,
(2) ?(?X) = ???(X) for all X ? g and ?? C,
(3) ?(X +Y) = ?(X) +?(Y) for all X,Y ? g, and
(4) ?[X,Y] = [?X,?Y] for all X,Y ? g.
Because of (2), ? is not an automorphism of g, but it is an automorphism of the real algebra
gR.
On the other hand, if ? : g ? g is a bijection such that (1)-(4) above hold, then 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.
16
LetB0,B, andBR denote the Killing forms of the Lie algebras g0, g, and gR, respectively.
Then [13, p.180]
B0(X,Y) = B(X,Y), X,Y ? g0
BR(X,Y) = 2ReB(X,Y), X,Y ? gR.
Since semisimplicity is equivalent to nondegeneracy of the Killing form, g0, g, and gR are all
semisimple if any of them is.
Every complex semisimple Lie algebra g has a compact real form g0, i.e., the group
generated by g0 is compact [13, p.181]. The compact real forms of complex simple Lie
algebras are listed in [13, p.516].
Example 3.1. For g = sln(C), the most obvious real form is sln(R); it is clear that gR =
sln(R)?isln(R). The corresponding conjugation is X 7? ?X. We note, however, that sln(R)
is not a compact real form for g.
Example 3.2. We construct a compact real form for sln(C) as follows: if X ? sln(C), set
X1 = 12(X ?X?) ? sln(C) and X2 = 12(X +X?) ? sln(C),
where ? denotes the conjugate transpose. In particular, X?1 = ?X1 and X?2 = X2 so that X1
and X2 are skew-hermitian and hermitian, respectively. Then X = X1 +X2. The set of all
trace zero skew-hermitian matrices is denoted sun(C) ? sln(C); in particular, if A ? sln(C)
is hermitian, then iA? sun(C), and
gR = sun(C)?isun(C).
In this case, conjugation is given by X 7? ?X?. The analytic subgroup of SLn(C) generated
by sun(C), denoted SUn(C), consists of determinant 1 unitary matrices and is compact, so
that sun(C) is a compact real form of sln(C).
17
3.2 Cartan Decomposition
We are particularly interested in algebras over R. Thus we make the notation more
convenient by using g to denote a real semisimple Lie algebra, gC its complexi?cation, and
? the conjugation of gC with respect to g. A (vector space) 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 gC such that
?(u) ? u, k = g?u, 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), X,Y ? g
is symmetric and positive de?nite. The following theorem establishes a one-to-one corre-
spondence between Cartan decompositions of a real semisimple Lie algebra and its Cartan
involutions [13, p.184] [25, p.144].
Theorem 3.3. Let g be a real semisimple Lie algebra written as 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 (X ? k,Y ? p) 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.
18
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 .
If g is a complex semisimple Lie algebra and u is a compact real form of g, then gR = u?iu
is automatically a Cartan decomposition [13, p.185].
The local (algebra-level) decomposition lifts nicely to a global (group-level) Cartan
decomposition. The details are summarized below [13, p.252] [21, p.362].
Theorem 3.4. Let G be a real 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.
In particular G = KP and every element g ? G can be uniquely written as g = kp,
where k ?K, p?P.
Remark 3.5. If G is compact, G = K and the theorem is trivial.
For any k ? K, Adk leaves k invariant because k is the Lie algebra of K. Since
Adk ? Autg, Adk leaves invariant the subspace of g orthogonal to k, which is exactly p.
Adk also leaves B invariant. If X ? g, write X = Xk +Xp with Xk ? k and Xp ? p and we
see that
Adk(?(X)) = Ad(k)Xk ?Ad(k)Xp = ?(Ad(k)Xk) +?(Ad(k)Xp) = ?(Ad(k)X),
19
i.e., Adk commutes with ?. Hence Adk leaves B invariant as well.
Example 3.6. Let us work out a Cartan decomposition of SLn(C). Since sun(C) is a
compact real form of sln(C), the decomposition
sln(C) = sun ?isun
is a Cartan decomposition with k = sun(C), p = isun(C), and Cartan involution the same
as the conjugation for the form, i.e., ?(X) = ?X?. As noted above, k has analytic subgroup
K = SUn(C) ? SLn(C), and P = expp is the set of determinant 1 matrices g so that g? = g.
The map ? is given by ?(g) = (g?)?1.
3.3 Root Space Decomposition
Let g be a complex semisimple Lie algebra. An element X ? 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, called
toral subalgebras. These subalgebras are abelian [20, p.35].
The normalizer of a subalgebra a of g is
Ng(a) = {X ? g : adX(a) ? a};
it is the largest subalgebra of g which contains a and in which a is an ideal. The centralizer
of a in g is
Zg(a) = {X ? g : adX(a) = 0}.
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 precisely the maximal toral subalgebras of
g [20, p.80], and all Cartan subalgebras of g are conjugate under the adjoint group Intg of
inner automorphisms [20, p.82].
20
Let h be a Cartan subalgebra of g. Since h is abelian, adgh is a commuting family of
semisimple endomorphisms of g, thus is a simultaneously diagonalizable family. In other
words, g is the direct sum of the subspaces
g = {X ? g : [H,X] = ?(H)X for all H ? h},
where ? ranges over the dual h? of h. Note that g0 = h because h is self-normalizing. A
nonzero ? ? h? is called a root of g relative to h if g ?= 0. The set of all roots, denoted by
?, is call the root system of g relative to h. Thus we have the root space decomposition
g = h?
?
??
g .
The root system ? characterizes g completely.
The restriction of the Killing form on h is nondegenerate and is given by
B(H,H?) =
?
??
?(H)?(H?), H,H? ? h.
Consequently we can explicitly identify h with h?: each ? ? h? corresponds to a unique
H ? h with
?(H) = B(H,H ) for all H ? h.
Thus it induces a nondegenerate bilinear form ??,?? de?ned on h? by
??,?? = B(H ,H ), ?,? ? h?.
The following is a collection of some properties of the root space decomposition [20,
p.36?40]:
(1) ? is ?nite and spans h?.
21
(2) If ?,? ? ??{0} and ?+? ?= 0, then B(g ,g ) = 0.
(3) If ?? ?, then ??? ?, but no other scalar multiple of ? is a root.
(4) If ?? ?, then [g ,g? ] is one dimensional, with basis H .
(5) If ?? ?, then dimg = 1.
(6) If ?,? ? ?, then 2??,????,?? ? Z and ?? 2??,????,?? ?? ?.
3.4 Restricted Root Space Decomposition
In this section, we follow the treatment in [21]. Let g be a real semisimple Lie algebra
with Cartan decomposition g = k?p and corresponding Cartan involution ?, and let a be a
maximal abelian subspace of p. For any element ? of the dual space a? of a, set
g := {X ? g : ad(H)X = ?(H)X for all H ? a}.
Analogous to the complex case, if ??= 0 and g ?= 0, we call ? a restricted root of g, or
a root of g with respect to a. We use ? to denote the set of roots of (g,a).
Set m = Zk(a), i.e.,
m = {X ? k : ad(X)H = 0 for all H ? a},
and
g0 = Zg(a).
Since a is a maximal abelian subspace of p, if X ? p so that ad(H)X = 0 for all H ? a,
then X ? a; so g0 ?p = a.
Proposition 3.7. The restricted root system has the following properties:
1. g is the orthogonal direct sum g = g0 ?
?
??
g .
22
2. [g ,g ] ? g + .
3. ?(g ) = g? .
4. g0 = a?m.
For each restricted root ?, the set
P = {X ? a : ?(X) = 0}
is a subspace of a of codimension 1. The subspaces P (? ? ?) divide a into several open
convex cones, called Weyl chambers. Fix a Weyl chamber a+ and refer to it as the funda-
mental Weyl chamber. A root ? is positive if it is positive on a+. If ? is not a positive root,
then since ? is a linear functional and is nonzero on a+, it must be negative on a+; in this
case we call ? a negative root. Let ?+ denote the set of all positive roots with respect to
a+, and ?? the set of all negative roots. If ?? ?+ and X ? g , write
X = Xk +Xp
with Xk ? k and Xp ? p. Since
[k,p] ? p and [p,p] ? k,
for any H ? a we have
(adH)Xk = ?(H)Xp, (adH)Xp = ?(H)Xk,
which imply
(adH)2Xk = ?(H)2Xk,
(adH)2Xp = ?(H)2Xp,
23
and
?(X) = Xk ?Xp ? g? .
For ?? ?+, de?ne
k = {X ? k : (adH)2X = ?(H)2X for all H ? a},
p = {X ? p : (adH)2X = ?(H)2X for all H ? a}.
Example 3.8. Again viewing g = sln(C) as a Lie algebra over R, we construct its restricted
root space decomposition using the Cartan decomposition
sln(C) = sun(C)?isun(C),
with k = sun(C) and p = isun(C). We choose the maximal abelian subspace a of p to
be the traceless real diagonal matrices. The restricted root ?ij (i ?= j) corresponds to the
two-dimensional root space
g ij = {a1Eij +a2iEij : a1,a2 ? R},
where Eij indicates the n?n matrix with zeros in each entry except for a 1 in the i,j entry.
The centralizer m of a in k is the set of traceless diagonal matrices with purely imaginary
entries. As a commutes only with diagonal matrices, it is clear that g0 = a ? m. Since the
Eij (i?= j), together with {Eii?Ei+1;i+1}, where i 0},
N =
?
??
??
?
??C B
0 (C?1)?
?
??: C real unit upper triangular, CB? = BC?
?
??
??.
3.6 Weyl Groups
Let G be a real semisimple Lie group with Lie algebra g, with a chosen Iwasawa decom-
position g = k?a?n. Let M be the centralizer of a in K and M? the normalizer of a in K,
i.e.,
M = {k ?K : Ad(k)H = H for all H ? a},
M? = {k ?K : Ad(k)a ? a}.
Note that M and M? are also the centralizer and normalizer of A in K, respectively, and
that they are closed Lie subgroups of K. More importantly, M is a normal subgroup of
M?, and the quotient group M?/M is ?nite because M and M? have the same Lie algebra
m = Zk(a) [13, p.284]. The ?nite group W := W(G,A) = M?/M is called the (analytically
de?ned) Weyl group of G relative to A. For w = mwM ?W, the linear map
Ad(mw) : a ? a
does not depend on the choice of mw ? M? representing w. Therefore, w 7? Ad(mw) is
well-de?ned, and we may regard w ? W as the linear map Ad(mw) : a ? a and W as a
28
group of linear operators on a. In particular, it is a faithful representation of W on a: for if
Ad(ms)X = Ad(mt)X for all X ? a, then
ms exp(X)m?1s = mt exp(X)m?1t for all X ? a.
Since A = expa, we see that msam?1s = mtam?1t for all a ? A, i.e., ms and mt are in the
same coset. Thus the representation is injective.
For each root ?, the re?ection s about the hyperplane P with respect to the Killing
form B, is a linear map on a given by
s (H) = H ? 2?(H)?(H
)
H , for all H ? a,
where H is the element of a representing ?, i.e., ?(H) = B(H,H ) for all H ? a. The
group W(g,a) generated by {s : ? ? ?} 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 [21, p.383].
We record a few of the many remarkable properties of W in the following proposition,
which we will have several occasions to use:
Proposition 3.13. [23, p.112] The Weyl group W permutes the Weyl chambers in a simply
transitive fashion, i.e., if C1 and C2 are Weyl chambers, then there is s?W so that s(C1) =
C2, and if s?= eW, then for any chamber C, S(C) ?= C.
Example 3.14. Let us compute the Weyl group for SLn(C). We use the conventions chosen
above for a and m, with a the set of traceless (real) diagonal matrices and m the set of
traceless diagonal matrices with purely imaginary entries. The subgroups generated by a
and m are A, the subgroup of determinant 1 diagonal matrices with real positive entries, and
M, the subgroup of determinant 1 diagonal matrices with entries of the form ei , ? ? R.
29
Let ej be the n?1 column vector with 1 in the jth row and 0s in the remaining entries.
The normalizer M? := NK(a) is the set of matrices of the form
(
ei 1e (1) ei 2e (2) ... ei ne (n)
)
,
where ?i ? R and ? is a permutation. Then M?/M ?= Sn.
The action of W on a is given by permutation of the entries of X ? a; if
X = diag(a1,...,an),
then
?(X) = diag(a!(1),...,a!(n)), ? ?Sn.
Alternatively, we may identify a permutation ? ? Sn with the unique permutation matrix
(also written as ?) in SLn(C), where ?ei = e!(i). The matrix representation of ? under the
standard basis is
? = [e!(1),??? ,e!(n)].
Thus if g ? SLn(C) is written in column form, g = [g1,??? ,gn], g? = [g!(1),??? ,g!(n)].
Moreover, if a = diag(x1,...,xn) ?A, then the action of ? on a is given by
??1diag(x1,...,xn)? = diag(x!(1),...,x!(n)). (3.2)
3.7 Bruhat Decomposition
Given a real semisimple Lie group G and s ? W, we again denote by ms ? M? a
representative such that s = msM. Moreover, for s = 1, we choose the identity of G for ms.
Because of the global Iwasawa decomposition (3.10), the exponential map is a di?eo-
morphism from a, n, and n? onto A, N, and N?, respectively. Thus each of A, N, and
N? is a closed subgroup of G [23, p.116]. Since M is also closed and centralizes A, and
30
MA normalizes N, MAN is a closed subgroup of G [13, p.403]. Thus multiplication from
M ?A?N to MAN is a di?eomorphism onto MAN [21, p.460].
Now by Proposition 3.13 there is s? ?W so that s?(a+) = ?a+. Then s??s?(a+) = a+,
so again by the proposition, s? = (s?)?1. Then it is clear that Adms (n?) = n?, so that
ms Nms = N?; (3.3)
alternatively, we write
ms N = N?ms . (3.4)
We refer to s? as the longest element of W.
The following theorem is one form of the Bruhat decomposition, which relies upon the
di?eomorphism outlined above and parameterizes G through W.
Theorem 3.15. [13, p.403] Let G be any noncompact semisimple Lie group. Then G is the
disjoint union
G =
?
s?W
MANmsMAN. (3.5)
Ifms = ms , the termMANmsMAN is an open submanifold ofG, and each termMANmtMAN
with t?= s? is a lower dimensional open submanifold.
Example 3.16. Let G = SLn(R), the group of determinant 1 matrices with entries from
R. We may choose N (N?) to consist of upper (respectively lower) triangular matrices with
diagonal entries all 1s; MAN is the set of upper triangular matrices, and W ?= Sn. Then
s? ?W has matrix representation
s? =
?
??
??
??
??
0 0 ... 0 1
0 0 ... 1 0
... ...
1 0 ... 0 0
?
??
??
??
??
.
31
Notice that s? = (s?)?1. It is clear that s?Ns? = N?: let n?N, i.e., of the form
?
??
??
??
??
??
??
1 ? ... ? ?
0 1 ... ? ?
... ... ...
0 0 ... 1 ?
0 0 ... 0 1
?
??
??
??
??
??
??
.
Since s? acts on the left by reversing the order of the columns, s?n has form
?
??
??
??
??
??
??
? ? ... ? 1
? ? ... 1 0
... ... ...
? 1 ... 0 0
1 0 ... 0 0
?
??
??
??
??
??
??
;
?nally s? acts on the right by reversing the order of the rows, so we see that s?ns? has form
?
??
??
??
??
??
??
1 0 ... 0 0
? 1 ... 0 0
... ... ...
? ? ... 1 0
? ? ... ? 1
?
??
??
??
??
??
??
.
In particular, (s?ns?)ij = nji, and we see that s?Ns? = N?, as claimed.
For the purposes of this dissertation, as well as for the classical LU decomposition
for GLn(C), the decomposition in 3.15 is not the most convenient form. The treatment
in [23] yields an alternate (but equivalent, up to the statements on open submanifolds)
decomposition.
32
Recall that M commutes with A, and M and A normalize N and N?. Since M?
normalizes M and A, ms normalizes M and A for any s ? W. Accordingly, we rewrite the
terms from Theorem 3.15:
MANmsMAN = NMAmsMAN = NmsMAN.
In addition, we note that msG = G for all s?W so that for any s?W,
G =
?
t?W
MANmtMAN =
?
t?W
msNmtMAN.
[23, p.118]. By (3.4), we have
ms NmtMAN = N?ms mtMAN
for any t?W and
{N?ms mtMAN : t?W} = {N?mtMAN : t?W}.
Thus the decomposition in Theorem 3.15 is equivalent to the form below.
Theorem 3.17. [23, p. 117] The real semisimple Lie group G has Bruhat decomposition
G =
?
s?W
N?msMAN, (3.6)
which is a disjoint union. Moreover, N?MAN is a di?eomorphic product and is an open
subset of G, and the other cells N?msMAN (s ?= eW) are lower dimensional submanifolds
of G.
An immediate consequence is that, for each g ? G, there exists a unique s ? W such
that g ?N?msMAN.
33
Given a decomposition of g ? G according to (3.6), say g = ?nmsman, we may use the
discussion preceding Theorem 3.17 to revert to the form (3.5) in Theorem 3.15; following
[18],
ms g = (ms ?nms )ms sman
?Nms sMAN
?MANms sMAN.
Thus the form (3.5) allows more ?exibility on the choice of the MA component. How-
ever, the form (3.6) is more convenient for the ensuing calculations, thus we shall use it as
the standard Bruhat decomposition in the remainder of this disseration.
Example 3.18. Let us consider the Bruhat decomposition of SLn(C). We slightly extend
our discussion from the semisimple SLn(C) to GLn(C), and the arguments from Example
3.14 are still applicable. Let {e1,...,en} be the standard basis of Cn, i.e., ei is a column
vector with 1 as the only nonzero entry in the i-th position. Given a matrix A ? Cn?n,
let A(i|j) denote the submatrix formed by the ?rst i rows and the ?rst j columns of A,
1 ? i,j ? n. The Bruhat decomposition of SLn(C) is indeed reduced to the well-known
Gelfand-Naimark decomposition [13, p.434].
Theorem 3.19. [16] Each A ? GLn(C) has A = L?U, for a permutation matrix ?, a
unit lower triangular matrix L ? GLn(C), and an upper triangular U ? GLn(C). The
permutation matrix ? is uniquely determined by A:
rank?(i|j) = rankA(i|j) for 1 ?i,j ?n.
Moreover diagU is uniquely determined by A.
Remark 3.20. Although ? is unique in the Gelfand-Naimark decomposition A = L?U of
A, the components L and U may be not unique. The following example illustrates this
34
ambiguity:
?
??0 1
1 1
?
??=
?
??1 0
1 1
?
??
?
??0 1
1 0
?
??
?
??1 0
0 1
?
??=
?
??1 0
0 1
?
??
?
??0 1
1 0
?
??
?
??1 1
0 1
?
??.
In contrast, the permutation ?? in a Gauss elimination A = ??L?U? may be not unique, but
L? and U? are uniquely determined by ??. For example,
?
??1 1
1 2
?
??=
?
??1 0
0 1
?
??
?
??1 0
1 1
?
??
?
??1 1
0 1
?
??=
?
??0 1
1 0
?
??
?
??1 0
1 1
?
??
?
??1 2
0 ?1
?
??.
Moreover, the ? in a Gelfand-Naimark decomposition A = L?U of A can also be a permuta-
tion in a Gauss elimination A = ?L?U? of A. To see this, we notice that ??1A = (??1L?)U
and det[(??1L?)(k|k)] = 1 since (??1L?)(k|k) is the submatrix formed by choosing the
?(1),??? ,?(k) rows and columns of L. Therefore, by the LU algorithm [19], ??1L? = L1U1
for some unit lower triangular L1 and unit upper triangular U1, and
A = L?U = ?(??1L?)U = ?L1(U1U) = ?L?U?, (3.7)
where L? := L1 and U? := U1U. We also have ??1A = L1U1U. Then u(A) can be computed
by
det[(??1A)(k|k)] = det[(L1U1U)(k|k)] = det[U(k|k)] =
k?
i=1
uii. (3.8)
Remark 3.21. When ? is the identity, it is well-known [19] that the decomposition A = LU
is unique.
3.8 Complete Multiplicative Jordan Decomposition
Let G be a real Lie group with Lie algebra g. An element g ? G is elliptic if Ad(g) ?
Autg is diagonalizable over C with eigenvalues of modulus 1; an element g ?G is hyperbolic
35
if g = expX, where X ? g is real semisimple, which is to say adX ? Endg is diagonalizable
over C with all eigenvalues real; an element g ?G is unipotent if g = expX, where X ? g is
nilpotent (3.3).
Each g ? G can be uniquely written as g = ehu, where e is elliptic, h is hyperbolic,
u is unipotent, and the three elements e, h, u commute [22, Proposition 2.1]. This is the
complete multiplicative Jordan decomposition (CMJD) of g. We write
g = e(g) h(g) u(g).
Let a+ be the closure of the Weyl chamber a+, and letA+ be the closure ofA+ := expa+.
Since exp : a ? A is a di?eomorphism onto, we have A+ = expa+ [23]. In addition,
di?eomorphisms preserve open sets so that A+ is open in A. It turns out that h ? G is
hyperbolic if and only if it is conjugate to an element of A+; in this case, such an element
of A+ is uniquely determined and we denote it by b(h) [22, Proposition 2.4]. For g ?G, we
de?ne
b(g) := b(h(g)) ?A+.
Example 3.22. Follow [13, Lemma 7.1]: viewing g ? SLn(R) as an element in gln(R), the
additive Jordan decomposition [19, p.153] for gln(R) yields
g = s+n1
(where s ? SLn(R) is semisimple, that is, diagonalizable over C, n1 ? sln(R) is nilpotent,
and sn1 = n1s). Moreover these conditions determine s and n1 completely [20, Proposition
4.2]. Put u := 1 +s?1n1 ? SLn(R) and we have the multiplicative Jordan decomposition
g = su,
36
wheresis semisimple, uis unipotent, andsu = us. By the uniqueness of the additive Jordan
decomposition, s and u are also completely determined. Since s is diagonalizable,
s = eh,
where e is elliptic, h is hyperbolic, eh = he, and these conditions completely determine e
and h. The decomposition can be obtained by observing that there is k ? SLn(C) such that
k?1sk = s1Ir1 ?????smIrm,
where s1 = ei 1|s1|,...,sm = ei m|sm| are the distinct eigenvalues of s with multiplicities
r1,...,rm respectively. Set
e := k(ei 1Ir1 ?????ei mIrm)k?1, h := k(|s1|Ir1 ?????|sm|Irm)k?1.
Since
ehu = g = ugu?1 = ueu?1uhu?1u,
the uniqueness of s, u, e and h implies e, u and h commute. Since g is ?xed under complex
conjugation, the uniqueness of e, h and u imply e,h,u ? SLn(R) [13, p.431]. Thus g = ehu
is the CMJD for SLn(R). The eigenvalues of h are simply the eigenvalue moduli of s and
thus of g.
A matrix in GLn(C) is called elliptic (respectively hyperbolic) if it is diagonalizable with
norm 1 (respectively real positive) eigenvalues. It is called unipotent if all its eigenvalues are
1. The complete multiplicative Jordan decomposition of g ? GLn(C) asserts that g = ehu for
e,h,u? GLn(C), where e is elliptic, h is hyperbolic, u is unipotent, and these three elements
commute. The decomposition is obvious when g is in a Jordan canonical form with diagonal
37
entries (i.e., eigenvalues) z1,??? ,zn, in which
e = diag
( z
1
|z1|,??? ,
zn
|zn|
)
, h = diag (|z1|,??? ,|zn|),
and u = h?1e?1g is a unit upper triangular matrix.
38
Chapter 4
Aluthge Iteration
Given 0 |?2|>???>|?n| (> 0). (5.1)
Let A = Xdiag(?1,...,?n)X?1. Assume that X and X?1 admit LU decomposition. Then
the sequence {As}s?N converges and the limit lims??As is an upper triangular matrix R in
which diagR = diag(?1,...,?n).
J.H. Wilkinson praised the LR iteration as ?algorithmic genius? [33, p.vii] and ?the
most signi?cant advance which has been made in connection with the eigenvalue problem
45
since the advent of automatic computers? with the understanding that ?The QR algorithm,
which was developed later by Francis, is closely related to the LR algorithm but is based
on the use of unitary transformations. In many respects this has proved to be the most
e?ective of known methods for the solution of the general algebraic eigenvalues problem.?
[33, p.487-488]. See the comments in [12].
Indeed Theorem 5.1 may be generalized due to the argument in [33, p.521-522]; with
careful reading (see Remark 5.10), we deduce the theorem below.
Theorem 5.2. Let A ? GLn(C) such that As admits LU decomposition for all s ? N and
the moduli of the eigenvalues ?1,...,?n of A are distinct. Let A = Xdiag(?1,...,?n)X?1.
Assume that PY = LYUY (Y := X?1) and XP? = LXUX admit LU decomposition
where P is a permutation matrix corresponding to the permutation ?. Then the sequence
{As}s?N converges and lims??As is an upper triangular matrix R in which diagR =
diag(?!(1),...,?!(n)).
The QR algorithm does not imply that its iterates converge to an upper triangular ma-
trix. However, it does assert that, under the assumption of distinct eigenvalue moduli of
A ? GLn(C), its iterates are ?convergent? to an upper triangular form. The LR algorithm,
on the other hand, does imply convergence, not just form convergence, towards an upper tri-
angular matrix R whose diagonal entries display the eigenvalues of the original matrix. Very
recently the QR algorithm has been extended in the context of semisimple Lie groups [18];
see Remark 5.5 for the comparison. We now extend the LR algorithm in the same fashion,
and give explicit examples. We remark that the decomposition PY = LYUY is the matrix
version of Gaussian elimination; none of the components P,LY,UY in the decomposition is
unique. However, the Gelfand-Naimark decomposition Y = L?U yields a unique ?, where L
is unit lower triangular and U is upper triangular. The role of the Gelfand-Naimark decom-
position will be played by the Bruhat decomposition in the context of semisimple groups, as
we will see in the next section.
46
5.2 Regular elements
Let G denote a connected real semisimple Lie group with a ?xed Cartan decomposition
g = k + p of the semisimple Lie algebra g. Let K ? G be the connected subgroup corre-
sponding to k. Recall that K is closed. Since AdK ? O(g), which is compact, AdK is itself
a maximal compact subgroup of AdG [13, Lemma 1.2, p.253].
Recall that a+ is the closure of the Weyl chamber a+, that A+ := expa+, and that
A+ = expa+ is the closure of A+. An element b ? A+ is regular if ?(logb) > 0 for all
positive roots ?, that is, b is in A+.
When G = SLn(C) or SLn(R), the CMJD of g ? G is given in Example 3.22; see [13,
p. 430-431]. In this case, b(g) is regular if and only if g has distinct eigenvalue moduli,
which implies that g is diagonalizable, that is, the unipotent part u(g) = 1. The following
proposition is an extension of this result in the context of a connected real semisimple Lie
group G.
Proposition 5.3. [18] Let g ? G such that b(g) ? A+ is regular. Then the unipotent
componentu(g) in the CMJD ofgis the identity and there isx?Gsuch thatxh(g)x?1 = b(g)
and xe(g)x?1 ?M.
Let ?+ denote the set of positive roots with respect to a+, and ?? the set of negative
roots. For any root ?, recall that g is the associated root space, and that
n? :=
?
??
g .
Given H ? a+, set
n0H :=
?
?P0H
g , where P0H := {?? ?? : ?(H) = 0},
47
and
nH :=
?
?PH
g , where PH := {?? ?? : ?(H) < 0}.
Thus n? decomposes as n? = n0H ?nH. De?ne the map
?0H : n? ? n0H
by projection onto the ?rst summand. If H ? a+, then P0H = ?, n0H = 0, and ?0H maps n? to
0.
Lemma 5.4. [18] Let b ? A+ and ? ? N?. Denote H := logb ? a+ and L := log? ? n?.
Then
lim
i??
bi?b?i = exp?0H(L) ?N?.
In particular, if b is regular, then lim
i??
bi?b?i = 1.
Proof. Let i> 0, and set Li := ead(iH)(L). By (2.2) and (2.3),
bi?b?i = exp(Ad(bi)(L))
= exp(Ad(eiH)(L))
= exp(ead(iH)(L))
= expLi.
For any root ? and L ? g , we see that
ad(iH)(L ) = iad(H)(L ) = i?(H)(L ).
Then
ead(iH)(L ) =
??
j=0
(ad(iH))j
j! (L ) =
??
j=0
(i?(H))j
j! L = e
i (H)L .
48
Now L? n?, so that it may be decomposed as
L =
?
??
L
with L ? g . Then
Li = ead(iH)(L) =
?
??
ei (H)L .
Now for all ?? ??, either ?(H) = 0 or ?(H) < 0 so that
lim
i??
bi?b?i = lim
i??
expLi = exp lim
i??
Li = exp?0H(L).
Lemma 5.5. [18] Let {xi}i?N and {yi}i?N be two sequences in G, such that limi??xi = 1
and {Ad(yi)}i?N is in a compact subset of Ad(G). Then
lim
i??
yixiy?1i = 1.
Proof. Recall that exp : g ?G is a di?eomorphism in a neighborhood of the identity. Since
limi??xi = 1, there exist N ? Z+ and Xi ? g (for i > N) so that if i > N, xi = expXi.
Thus limi??Xi = 0, and for all i>N, yixiy?1i = exp(Ad(yi)Xi).
By way of contradiction, suppose that limi??yixiy?1i ?= 1. Then limi??(Ad(yi)Xi) ?=
0, so that there is an open neighborhood U ? g of 0 and a subsequence {ti}i?N of N so that
ti >N and Ad(yti)Xti ??U for all i? N. However, since {Ad(yti)} is in a compact subset of
AdG, there is a subsequence {si}i?N of {ti}i?N and y ? G so that limi??ysi = Ady. Thus
limi??Ad(ysi)Xsi = Ad(y)(0) = 0, which contradicts the assumption Ad(yti)Xti ?? U for
all i? N. Thus limi??yixiy?1i = 1.
Proposition 5.6. Given ? ?W, set G! := N?m!MAN. Then m?1! G! ?N?MAN.
49
Proof. Let g := ?nm!man?G!, with ?n?N?, m?M, a?A, and n?N. By [23, p.117]
m!N?m?1! = N1N2
is a di?eomorphic product, where N1 and N2 are Lie subgroups of N? and N, respectively.
So
m?1! ?n?1m! = ?n?n?
with ?n? ?N? and n?N. Then
m?1! ?nm!man = ?n?n?man = ?n?man?? (5.2)
with n?? ? N since MA normalizes N. Thus (5.2) is the Bruhat decomposition of m?1! g ?
N?MAN.
5.3 Bruhat iteration
Let g = g1 = ?nman be the Bruhat decomposition of g ?N?MAN, where ?n?N? and
man?MAN. Set ?n1 := ?n and u1 := man so that g1 = ?n1u1. De?ne
B(g1) := u1?n1 = ?n?11 g1?n1
and the Bruhat iteration recursively:
gs = B(gs?1) = ?n?1s?1gs?1?ns?1, s? N.
Theorem 5.7. Suppose that g ?G with b(g) regular, {gs}s?N ?N?MAN, and g = xcbx?1
as in Proposition 5.3. If x?1 = lm!p, l ? N?, ? ? W, p ? MAN, and xm! ? N?MAN,
50
then gs ?N?MAN. Moreover, there is p? ?MAN so that
lims??gs = p?m?1! (bc)m!(p?)?1 ?MAN.
The MA-component of the limit is ??(bc).
Proof. Note that
gs = ?n?1s?1gs?1?ns?1 = ?n?1s?1????n?11 g1?n1????ns?1. (5.3)
Set ts := ?n1????ns and rs := us???u1. Then
ts?1gs = g1ts?1. (5.4)
Now consider
tsrs = ?n1????ns?1(?nsus)us?1???u1
= ?n1????ns?1gsus?1???u1
= g1?n1????ns?1us?1???u1
= g1ts?1us?1.
Repeated application of this result yields tsus = gs1, i.e., gs1 ?N?MAN for all s?N.
As ts ?N? and rs ?MAN the Bruhat decomposition of gs1 is given by
gs1 = tsrs. (5.5)
Let g1 = ehu be the complete multiplicative Jordan decomposition of g1. As g1 is regular,
there is an x ? G so that b := xhx?1 ? A+. By [18], u = 1 and c := xex?1 ? M, so that
x?1g1x = cb?MA+.
51
Suppose that x?1 = lm!p, l ?N?, ? ?W, and p?MAN. We want to show that
lims??csbslb?sc?s = 1.
Since c?M ?K, {Adcs}s?N is contained in the compact set AdK. By Lemma 5.5 we need
only consider the sequence {bslb?s}s?N. By Lemma 5.4,
lims??bslb?s = 1
since b is regular. Set
ls:=csbslb?sc?s, (5.6)
which is in n? since MA normalizes N?. Now
gs = xcsbsx?1 = xcsbsl(csbs)?1csbsm!p = xlscsbsm!p. (5.7)
Since M? normalizes MA, csbsm! = m!csbs for some cs ?M, bs ?A. We have
xlscsbsm!p = xlsm!csbsp,
with ks := csbsp?MAN. Then (5.7) can be rewritten as
gs =xlsm!csbsp = xlsm!ks. (5.8)
Since ls ? N? and ks ? MAN, lsm!ks is in the cell G!. Then by Proposition 5.2
m?1! lsm!ks ?N?MAN, i.e.,
m?1! lsm!ks = l?sk?s
with l?s ? N? and k?s ? MAN. By Proposition 5.6, m?1! lsm! = ?lsps, with ?ls ? N? and
ps ? N. Since ls ? 1, m?1! lsm! = ?lsps ? 1 as well. By the di?eomorphic product of
52
N?MAN, both ?ls and ps approach 1. Thus we rewrite (5.8) as
gs = xlsm!ks = xm!(m?1! lsm!)ks = xm!?lspsks.
By the assumption, xm! ? N?MAN, say xm! = l?p?. We have xm!?lspsks = l?p??lspsks.
Since l? ?N?, p? ?MAN, ?ls ?N? with ?ls ? 1, and ps ? 1, we have
tsrsk?1s ?l?p?.
Since the mapN??M?A?N?N?MAN is a di?eomorphism, andts ?N?,rsk?1s ?MAN,
tsrsk?1s ? l?p? implies that ts ? l? and rsk?1s ? p? as s ? ?. Since gs = t?1s?1g1ts?1 from
(5.4), we have
gs ? (l?)?1g1l? = (xm!(p?)?1)?1g1(xm!(p?)?1) = p?m?1! (bc)m!p?.
We record as a corollary the special case when ? = 1, i.e., x?1 is in the large cell.
Corollary 5.8. Suppose that g ?G with b(g) regular, {gs}s?N ?N?MAN, and g = xcbx?1
as in Proposition 5.3. If x?1 and x ? N?MAN, then gs ? N?MAN. Moreover, there is
p? ?MAN so that
lims??gs = p?(bc)(p?)?1 ?N.
The MA-component of the limit is bc.
Remark 5.9. We remark that the large cell N?MAN is not closed under inversion. For
example, for G = SL2(C), note that and
A = LU =
?
?? 1 0
?1 1
?
??
?
??1 1
0 1
?
??=
?
?? 1 1
?1 0
?
??
53
but
(LU)?1 =
?
?? 0 1
?1 1
?
??,
which is not in the large cell.
Remark 5.10. The proofs in the literature of the convergence statements for the LR itera-
tion are not purely group theoretic since they usually make use of the embedding of GLn(C)
in Cn?n to accommodate matrix addition at some point (see [33, p.521] for example). Two
proofs of the convergence statement are given in [33]. The ?rst is computation-intensive,
while the second is much more similar to the proof given here, except that it uses Gaussian
elimination instead of Gelfand-Naimark decomposition. However, both proofs are sketched
out roughly, and are missing important details, for example, Equation (33.2) in [33, p.521].
Our proof of the generalization only involves purely group theoretic arguments.
Remark 5.11. For x?1 ? N?m!MAN, the condition xm! ? N?MAN is essential to
guarantee convergence in Theorem 5.7. Consider Rutishauser?s matrix [26, p.52]
A1 :=
?
??
??
?
1 ?1 1
4 6 ?1
4 4 1
?
??
??
?
=
?
??
??
?
0 ?1 ?1
1 1 1
1 0 1
?
??
??
?
?
??
??
?
5 0 0
0 2 0
0 0 1
?
??
??
?
?
??
??
?
1 1 0
0 1 ?1
?1 ?1 1
?
??
??
?
.
Setting
x :=
?
??
??
?
0 ?1 ?1
1 1 1
1 0 1
?
??
??
?
,
we see that x?1 has LU decomposition
x?1 =
?
??
??
?
1 0 0
0 1 0
?1 0 1
?
??
??
?
?
??
??
?
1 1 0
0 1 ?1
0 0 1
?
??
??
?
,
54
i.e., x?1 is in the largest cell, while x is not in the largest cell.
We claim that the sequence As remains in the largest cell, but diverges. To this end we
?rst establish that
As+1 =
?
??
??
?
1 ?15s 1
4?5s 6 ?5s
4 45s 1
?
??
??
?
by induction. A routine calculation shows that
A2 =
?
??
??
?
1 ?15 1
20 6 ?5
4 45 1
?
??
??
?
.
Suppose that
As =
?
??
??
?
1 ?15s 1 1
4?5s?1 6 ?5s?1
1 ?15s 1 1
?
??
??
?
.
Then As has LU decomposition
As =
?
??
??
?
1 0 0
4?5s?1 1 0
4 45s 1
?
??
??
?
?
??
??
?
1 ?15s 1 1
0 10 ?5s
0 0 1
?
??
??
?
.
So
As+1 =
?
??
??
?
1 ?15s 1 1
0 10 ?5s
0 0 1
?
??
??
?
?
??
??
?
1 0 0
4?5s?1 1 0
4 45s 1
?
??
??
?
=
?
??
??
?
1 ?15s 1
4?5s 6 ?5s
4 45s 1
?
??
??
?
.
The sequence obviously diverges since 4?5s ? ?.
Remark 5.12. In Theorem 5.2, the condition is given in terms of Gaussian eliminationPY =
LYUY instead of Gelfand-Naimark decomposition Y = L?U. See [16] for some comparison
55
of the two decompositions. Although ? is unique in the Gelfand-Naimark decomposition
Y = L?U, the components L and U may be not unique. The permutation P may be not
unique, but LY and UY are uniquely determined by the permutation matrix P. Moreover,
? can also be a permutation in a Gauss elimination Y = ?L?U? [16].
Example 5.13. Consider the real symplectic group ([28, p.129]):
G := Spn(R) = {g ? SL2n(R) : g?Jng = Jn}, Jn =
?
?? 0 In
?In 0
?
??.
Recall that the elements of G are of the form
?
??A B
C D
?
??, A?C = C?A, B?D = D?B, A?D?C?B = I
n (5.9)
[28, p.128], and that the Cartan decomposition of G is given by
K =
{
?
?? C B
?B C
?
??: C +iB ? U(n)}= O(2n)?Sp
n(R),
A = {diag(a1,...,an,a?11 ,...,a?1n ) : a1,...,an > 0},
N =
{
?
??C B
0 (C?1)?
?
??: C unit upper triangular, CB? = BC?}
The centralizer M of A in K is the group of the diagonal matrices in K, i.e., the group of
matrices of the form diag(C,C), where C = diag(?1,?1,...,?1) (independent signs here
and below). The normalizer M? of A in K is W?M where W? is generated by
{Ek;n+k ?En+k;k +?i?=k;n+kEii : k = 1,...,n}
?{diag(C,C) : C is a permutation matrix}.
56
Note that W?M/M ? W?/(W? ?M) is isomorphic to the Weyl group. In particular, W ?=
(Z/2Z)n oSn [20, p.66]. We have N? = N? = {n? : n?N}.
We may choose a to be the set of all real matrices of the form
X = diag(x1,...xn,?x1,...,?xn) ?= (x1,...,xn);
the natural basis for a is then
{Hi := Ei;i ?En+i;n+i : 1 ?i?n}.
The corresponding basis elements Li of a? are given by Li(Hj) = ?ij. The Lie algebra n of
N is
n =
{
?
??A B
0 ?A?
?
??: A,B ? gl
n(R),A stricly upper triangular,B = B
?
}
.
The root system is {?Li ?Lj : i ?= j}?{2Li : 1 ? i ? n}; the positive roots ? are
{Li ?Lj,Li +Lj,2Li : ib2 > ??? >bn > 1. Since bc = cb, it
follows that c = diag(C,C) with C = diag(?1,?1,...,?1).
According to Theorem 5.7,
1. lim
i??
k(gi) = cs ? M, where cs := (msm)?1c(msm) is of the form diag(C,C) with
C = diag(?1,?1,...,?1),
2. lim
i??
a(gi) = bs ?A, wherebs := (msm)?1b(msm) = m?1s bms is of the form diag(D,D?1)
with D a diagonal matrix having diagonal entries b?11 ,b?12 ,...,b?1n in some order.
The diagonal entries of n(gi) are each 1, so it follows that the diagonal entries of the sequence
{gi} converge to the eigenvalues of g.
5.4 Open Problem
One often encounters real matrices A ? SLn(R) whose complex eigenvalues occur in
complex conjugate pairs, i.e., the hyperbolic component of A is not regular. In order to deal
with this case, we would like to relax somewhat the assumption of regularity in Theorem
5.7. In particular, we would like to consider g ? G so that xgx?1 = cb for some y ? G,
c?K, b?A+, such that cb = bc. In SLn(C), this corresponds to choosing a matrix X with
repeated eigenvalues. In this case, we believe that the convergence pattern for X will be
based upon the repeated eigenvalues; if the distinct eigenvalues are ?1 >...>?j, so that ?i
58
has multiplicity ki, then we expect that X (pattern) converges to the block diagonal form
?
??
??
??
??
A 1 ? ... ?
0 A 2 ?
...
0 0 ... A j
?
??
??
??
??
,
where A i is a ki ?ki matrix with eigenvalues all ?i. This requires further study.
5.5 Comparison of the Iwasawa and Bruhat Iterations
Given X ? GLn(C), the QR algorithm writes X as a product of a unitary matrix Q
and upper triangular matrix R,
X = QR.
We de?ne the QR iteration in a similar fashion to the LR iteration by setting
X1 = X = Q1R1,
and de?ning:
Xs+1 = RsQs = Qs+1Rs+1, s = 1,2...
The iteration preserves the eigenvalues of X1 since Xs+1 = RsXsR?1s . Indeed, if X1
has distinct eigenvalue moduli, proofs in the literature [17, 35] show that {Xi}i?N displays
pattern convergence to a matrix in upper triangular form. In particular, {Xi}i?N does not
necessarily converge, as the strictly upper triangular entries may behave poorly. However,
the diagonal entries will actually converge to the eigenvalues of X1.
59
TheQRiteration has the advantage of being computationally stable, but as it only guar-
antees form convergence, not actual convergence, it compares poorly with the LR iteration,
which does indeed guarantee actual convergence.
The Iwasawa decomposition of SLn(C) corresponds to the QR decomposition, thus pro-
vides motivation for considering a generalized QR iteration in this context. The Iwasawa
iteration of g ? G, where G is a real connected semisimple Lie group with Iwasawa decom-
position G = KAN, is de?ned by
g1 := g = k1a1n1,
and
gs+1 = asnsks = ks+1as+1ns+1, s = 1,2,...
The asymptotic behavior of this sequence is given in the following theorem:
Theorem 5.14. [18] Let g ?G and assume that ygy?1 = cb for some y ?G, c?K, b?A+,
so that cb = bc. Suppose that y has Bruhat decomposition
y = n?msman?N?msMAN.
Let n?0 := exp?0H(L) where H := logb? a+ and L := logn? ? n?. Put
cs := (n?0 msm)?1cs(n?0 msm).
Then there exists a sequence {di}i?N in the set AN?csAN ?K such that
lim
i??
kid?1i = 1.
60
In keeping with the matrix result, the theorem does not suggest actual convergence of
the Iwasawa iteration. Instead, it speci?es the behavior of theK component of the sequence:
it ?nearly converges? in the sense that there is a ?multiplier sequence?{di}which can be used
to perturb {ki} into a converging sequence. Again, the Bruhat iteration is advantageous over
the Iwasawa iteration in that it can guarantee actual convergence, not just form convergence.
61
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