Toeplitz Matrices are Unitarily Similar to Symmetric Matrices

Abstract

We prove that Toeplitz matrices are unitarily similar to complex symmetric matrices. Moreover, two $n\times n$ unitary matrices that uniformly turn all $n\times n$ Toeplitz matrices via similarity to complex symmetric matrices are explicitly given, respectively. When $n\leq 3$, we prove that each complex symmetric matrix is unitarily similar to some Toeplitz matrix, but the statement is false when n > 3.