V. N. Chugunov

Classification of real pairs of commuting Toeplitz and Hankel matrices

In this paper, a complete description of the set of matrix pairs (T,H) such that T is a real Toeplitz matrix, H is a real Hankel matrix, and TH = HT is given.

On conditions for permutability of Toeplitz and Hankel matrices

The problem of describing pairs of commuting matrices (T, H), where T and H are a Toeplitz and a Hankel matrix, respectively, is examined. Several families of such pairs are indicated.

On eigenvalues of (T+H)-circulants and (T+H)-skew-circulants

Explicit formulas for calculating eigenvalues of the Hankel circulants, Hankel skewcirculants, (T + H)-circulants, and (T + H)-skew-circulants are obtained. It is shown that, if ϕ ≠ ±1, then the set of matrices that can be represented as sums of a Toeplitz ϕ-circulant and a Hankel ϕ-circulant is not an algebra.

Fast Algorithms for Calculating the Eigenvalues of Normal Hankel Matrices

Algorithms for the rapid computation of eigenvalues for certain classes of normal Toeplitz matrices are presented. The time required to execute these algorithms is compared to that of the standard Matlab procedure eig.

A Description of Pairs of Quasi-Commuting Toeplitz and Hankel Matrices

We say that square matrices A and B of the same order quasi-commute if AB = σBA for a certain scalar σ. The classical definitions of commutation and anti-commutation are particular cases of the above definition. We give a complete description of pairs of quasi-commuting Toeplitz and Hankel matrices for σ ≠ ±1.

Classifying anti-commuting pairs of Toeplitz and Hankel matrices

Conditions for commuting a Toeplitz matrix and a Hankel matrix were obtained relatively recently (in 2015). The solution to the problem of describing all anti-commuting pairs (T, H), where T is a Toeplitz matrix and H is a Hankel matrix, is sketched below.

A duality relation for unitary automorphisms in the spaces of Toeplitz and Hankel matrices

The duality relation in the title of the paper is an identity between the groups of unitary automorphisms acting in the space of Toeplitz or Hankel matrices by similarity or congruence. A simple answer is given to the question why such identities can emerge.

Unitary automorphisms of the space of Toeplitz-plus-Hankel matrices

Abstract Our motivation was a paper of 1991 indicating three special unitary matrices that map Hermitian Toeplitz matrices by similarity into real Toeplitz-plus-Hankel matrices. Generalizing this result, we give a complete description of unitary similarity automorphisms of the space of Toeplitz-plus-Hankel matrices.

Unitary automorphisms of the space of (T+H)-matrices of order four

Matrices U in unitary group U4 that satisfy the implication ∀Aℱℋ4 → B =U*AU ∈ ℱℋ4 are examined. Here, ℱℋ4 is a set of order four (T+H)-matrices. Such matrices U can be identified with unitary automorphisms of the space ℱℋ4.Our problem is whether the boundary of U can be free of zeros. (The boundary of a matrix is the collection of its entries in the first and last row and the first and last column.) It is shown that matrices U with an entirely nonzero boundary do exist, in contrast to the situation for the unitary automorphisms of the spaces of order four Toeplitz and Hankel matrices.

Classifying pairs of commuting Toeplitz and Hankel matrices

Conditions under which two Toeplitz matrices are permutable have been known since at least 1995. Recently (in 2011) V.I. Gel’fgat stated conditions for permutability of a pair of Hankel matrices. In this communication, we outline a scheme for solving the much more difficult problem of describing all the pairs (T, H) such that T is a Toeplitz matrix, H is a Hankel matrix, and T commutes with H.