Toeplitz-plus-Hankel circulants are reducible to block diagonal form via unitary congruences

Abstract

A Hankel circulant is a matrix obtained by reversing the order of columns (or rows) in a conventional circulant. A Toeplitz-plus-Hankel circulant (briefly, ()-circulant) is the sum of a circulant and a Hankel circulant. Bozzo discovered that the set of ()-circulants is the centralizer of the matrix , where is the cyclic permutation matrix. As a consequence, all the matrices in can be simultaneously brought to a block diagonal form with diagonal blocks of orders one and two by a unitary similarity transformation. We show that the same assertion holds for if unitary similarities are replaced by unitary congruences.

Keywords:

• circulant
• Hankel circulant
• T + H-circulant
• centralizer
• discrete Fourier transform
• unitary similarity transformation
• unitary congruence transformation