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If the cosquares of complex matrices $$A$$ and $$B$$ are similar and there is a unimodular number in the spectrum of the cosquares, then $$A$$ and $$B$$ are not necessarily congruent. Assume that such an eigenvalue $$\lambda_0$$ is unique. In this case, so far, one could verify the congruence of $$A$$ and $$B$$ by using a rational algorithm only in two situations: (1) the eigenvalue $$\lambda_0$$ is simple or semi-simple; (2)there is only one Jordan block associated with $$\lambda_0$$ in the Jordan form of the cosquares. We propose a rational algorithm for checking congruence in the case where two Jordan blocks of the same order are associated with $$\lambda_0$$ .

A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm for checking the congruence of arbitrary complex matrices A and B is known. The situation can be different if both A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. This paper proposes a rational algorithm for checking whether two involutive matrices A and B are congruent.

Let $$A$$ be a complex $$n \times n$$ matrix. The set $$\mathcal{L}$$ of matrices $$X$$ satisfying the relation $${{X}^{T}}AX = A$$ is called the $$T$$ -congruence centralizer of $$A$$ . It is shown that the calculation of matrices from the nonlinear variety $$\mathcal{L}$$ can be reduced to solving a linear matrix equation.

A known criterion of the congruence between nonsingular matrices ( $$A$$ and $$B$$ are congruent if and only if their cosquares $$A^{-T}A$$ and $$B^{-T}B$$ are similar) is interpreted from the group-theoretic viewpoint. This interpretation makes it possible to understand why it is the cosquares of $$A$$ and $$B$$ that participate in the above criterion.

A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm is available that is able to check the congruence of arbitrary complex matrices A and B. In this paper, we propose a rational algorithm of verification that works under less stringent conditions than all the conditions known by now.

A finite algorithm that uses arithmetic operations only is said to be rational. There exist rational methods for checking the congruence of a pair of Hermitian matrices or a pair of unitary ones. We propose a rational algorithm for checking the congruence of general normal matrices.

Complex n×n matrices A and B are said to be T-congruent if B = STAS and ∗-congruent if B = S∗AS, where S is an arbitrary nonsingular matrix. For several facts related to normal matrices and ∗-congruences, their counterparts in the theory of T-congruences concerning conjugate-normal matrices are found. Bibliography: 6 titles.

A finite computational process using arithmetic operations only is called a rational algorithm. Matrices A and F are said to be Hermitian congruent if F = Q∗AQ for a nonsingular matrix Q. The paper is a survey of the necessary conditions for Hermitian congruences that are verifiable by rational algorithms. Bibliography: 7 titles.

The paper discusses the possibility of reducing a square complex matrix A to a direct sum of smaller matrices by using ∗-congruence transformations. It turns out that this possibility is related to appropriate partitions of the spectrum of the cosquare of A. This makes it possible to associate the direct summands of the sum with subsets of the latter spectrum.

The paper describes the nonsingular matrices U such that for every Hankel matrix A of the same order, the matrix U−1AU also is a Hankel matrix. Bibliography: 4 titles.

Peculiarities of the Toeplitz and polar decompositions of an involutive matrix are described. For instance, it is shown that the unitary factor in both polar decompositions of such a matrix is itself an involution; consequently, it is not only a unitary matrix but a Hermitian matrix as well.

A finite computational process using only arithmetical operations is called a rational algorithm. Presently, there is no known rational algorithm for checking congruence between arbitrary complex matrices A and B. The situation may be different if A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. In this publication, we propose a rational algorithm for checking congruence between matrices A and B that are square roots of Hermitian matrices.

The following result is attributed to J. Williamson: Every real, symmetric, and positive definite matrix A of even order n = 2m can be brought to diagonal form by a congruence transformation with symplectic matrix. The diagonal entries of this form are invariants of congruence transformations performed with A, and they are called the symplectic eigenvalues of this matrix. This short paper proves an analogous fact concerning (complex) skew-symmetric matrices and transformations belonging to a different group, namely, the group of pseudo-orthogonal matrices.

Let Tn be the set of complex Toeplitz n × n matrices. The paper describes the matrices U in the linear group GLn(ℂ) such that $$ \forall A\in {T}_n\to {U}^{-1} AU\in {T}_n $$ and also the matrices U ∈ GLn(ℂ) such that $$ \forall A\in {T}_n\to {U}^{-1}A\overline{U}\in {T}_n. $$

Let A be a nonsingular complex (n × n) matrix. The congruence centralizer of A is the collection $$\mathcal{L}$$ of matrices X satisfying the relation $$X{\kern 1pt} {\text{*}}AX = A$$ . The dimension of $$\mathcal{L}$$ as a real variety in the matrix space $${{M}_{n}}({\mathbf{C}})$$ is shown to be equal to the difference of the real dimensions of the following two sets: the conventional centralizer of the matrix $${{A}^{ - }}{\text{*}}A$$ , called the cosquare of A, and the matrix set described by the relation $$X = {{A}^{{ - 1}}}X{\kern 1pt} {\text{*}}A$$ . This dimensional formula is the complex analog of the classical result of A. Voss, which refers to another type of involution in $${{M}_{n}}({\mathbf{C}})$$ .

A number of localization regions that can be found by rational algorithms are indicated for the eigenvalues of a normal matrix. Rational algorithms are finite procedures using arithmetic operations only.

With a square complex matrix A the matrix pair consisting of its symmetric S(A) = (A + AT)/2 and skew-symmetric K(A) = (A − AT)/2 parts is associated. It is shown that square matrices A and B are congruent if and only if the associated pairs (S(A), K(A)) and (S(B), K(B)) are (strictly) equivalent. This criterion can be verified by a rational calculation, provided that the entries of A and B are rational or rational Gaussian numbers.

Let K be a nonsingular skew-symmetric matrix of even order n = 2m. For such a matrix, the paper proposes a finite algorithm for computing an m-dimensional neutral subspace, which uses arithmetic operations and quadratic radicals only. The necessity of computing neutral subspaces originates in the problem of solving quadratic matrix equations.

The concept of a regularizing decomposition was introduced by R. Horn and V. Sergeichuk. It means the representation of a square matrix by a direct sum of Jordan blocks with zero on the principal diagonal and a nonsingular matrix. Such a representation is attained via congruence transformations and differs from the Jordan normal form. For the reasons explained in this paper, we prefer to speak of an SR decomposition (in other words, a singular-regular decomposition) of a matrix rather than a regularizing decomposition. Accordingly, algorithms providing this decomposition are called SR algorithms.We develop a rational algorithm that considerably simplifies the SR algorithms proposed by Horn and Sergeichuk.

The concept of the minimal polyanalytic polynomial was introduced by M. Huhtanen in connection with the generalized Lanczos process as applied to a normal matrix. The paper discusses the possibility of finding an equivalent of the characteristic polynomial in the set of polyanalytic polynomials.

The CMV matrix is the five-diagonal matrix that represents the operator of multiplication by the independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle C. The article by Cantero, Moral, and Velazquez, published in 2003 and describing this matrix, has attracted much attention because it implies that the conventional orthogonal polynomials on C can be interpreted as the characteristic polynomials of the leading principal submatrices of a certain five-diagonal matrix. The present paper recalls that finite-dimensional sections of the CMV matrix appeared in papers on the unitary eigenvalue problem long before the article by Cantero et al. was published. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.

A complete description of the sets of pairs of anti-commuting Toeplitz and Hankel matrices is given.

C. R. Johnson and S. Furtado have shown that if two unitary matrices are *-congruent, then they are unitarily similar. A counterpart of this statement concerning another type of matrix congruence, namely, the T-congruence is established. In addition, the problem of checking the T-congruence of given matrices using a finite number of arithmetic operations is discussed.

It is shown that every solution of the matrix equation in the title of the paper, where n = 2m, can be transformed by a symmetric permutation of its rows and columns to a direct sum of two triangular Toeplitz matrices of order m.

As is known, the cosquare of an arbitrary nonsingular Hermitian matrix is the identity matrix, and the cosquare of an arbitrary unitary matrix is unitary. In a complex linear space with the symplectic metric, the symplectic and skew-Hamiltonian matrices are the counterparts of the unitary and Hermitian matrices, respectively. Specific features of cosquares for these two matrix classes are indicated.

A simple proof of Williamson’s theorem is given. This theorem states that a real symmetric positive definite matrix A of even order can be brought to diagonal form Λ by a symplectic congruence transformation. The diagonal entries of Λ are called symplectic eigenvalues of A. The problem of calculating these values is also discussed.

A technique that can be considered an analog of the bisection method for Hermitian matrices is proposed for general normal 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.

Algorithms of the Bartels–Stewart type for the numerical solution of Sylvester matrix equations of modest size are modified for the case where the linear operators associated with these equations are normal. The superiority of the modified algorithms over the original ones is illustrated by numerical results.

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.

Consider C n as the pseudo-unitary space with the inner product defined by the matrix $$ {p}_n=\left(\begin{array}{l}\kern4em 1\hfill \\ {}\kern3em 1\hfill \\ {}\kern1em \dots \hfill \\ {}1\hfill \end{array}\right) $$ . In this space, centro-unitary matrices play the role of unitary operators. The main result of this paper is a factorization of an arbitrary centro-unitary matrix of even order into a product of simpler centro-unitary matrices. This result is an implication of a similar fact concerning factorizations of pseudo-unitary matrices of the type (n, n).

Let a complex matrix A be the direct sum of its square submatrices B and C that have no common eigenvalues. Then every matrix X belonging to the centralizer of A has the same block diagonal form as the matrix A. This paper discusses how the conditions on the submatrices B and C should be modified for an analogous assertion about the congruent centralizer of A, which is the set of matrices X such that X * AX = A, to be valid. Also the question whether the matrices in the congruent centralizer are block diagonal if A is a block anti-diagonal matrix is considered. Bibliography: 2 titles.

The congruent centralizer of a complex n × n matrix A is the set of n × n matrices Z such that Z*AZ = A. This set is an analog of the classical centralizer in the case where the similarity relation on the space of n × n matrices is replaced by the congruence relation. The study of the classical centralizer C A reduces to describing the set of solutions of the linear matrix equation AZ = ZA. The structure of this set is well known and is explained in many monographs on matrix theory. As to the congruent centralizer, its analysis amounts to a description of the solution set of a system of n 2 quadratic equations for n 2 unknowns. The complexity of this problem is the reason why there is still no description of the congruent centralizer $$ {C}_J^{\ast } $$ even in the simplest case of the Jordan block J = J n (0) with zeros on the principal diagonal. This paper presents certain facts concerning the structure of matrices in $$ {C}_J^{\ast } $$ for an arbitrary n and then gives complete descriptions of the groups $$ {C}_J^{\ast } $$ for n = 2, 3, 4, 5.

The paper describes the congruent centralizer of the matrix $$ {\Delta}_n=\left(\begin{array}{llll}\hfill & \hfill & \hfill & 1\hfill \\ {}\hfill & \hfill & \dots \hfill & i\hfill \\ {}\hfill & 1\hfill & \dots \hfill & \hfill \\ {}1\hfill & i\hfill & \hfill & \hfill \end{array}\right) $$ , representing one of three blocks in the Horn–Sergeichuk canonical form, i.e., the set of matrices X such that X ∗Δ n X = Δ n .

A simple method is given for a priori determination of the number of Jordan blocks and their orders in the regularizing decomposition of a square matrix. This decomposition was proposed by R. Horn and V. Sergeichuk. It is attained by congruence transformations and makes it possible to separate the regular and singular parts of the matrix.

Solutions to the sesquilinear matrix equation X*DX + AX + X*B + C = 0, where all matrices are of size n × n, are put in correspondence with n-dimensional neutral (or isotropic) subspaces of the associated matrix M of order 2n. A way of constructing such subspaces is proposed for when M is a symmetric quasi-definite matrix of the (n, n) type.