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Let Ip,q = Ip ⊕−Iq. Pseudounitary eigenvalues of a positive definite matrix A are the moduli of conventional eigenvalues of the matrix Ip,qA. They are invariants of pseudounitary *-congruences performed with A. For a fixed n = p+q, the sum of the squares σp,q of these numbers is a function of the parameter p. In general, its values for different p can differ very significantly. However, if A is the tridiagonal Toeplitz matrix with an entry a ≥ 2 on the principal diagonal and the entry −1 on the two neighboring diagonals, then σp,q has the same value for all p. This nontrivial fact is explained in the paper. [ABSTRACT FROM AUTHOR]

Let 풜 be an algebra over a field 픽 generated by a set of matrices 풮. The paper considers algorithmic aspects of checking whether 풜 coincides with the full matrix algebra. Laffey has shown that for 픽 = ℂ, under the assumption that 풮 contains a Jordan matrix from a certain class, there is a fast method for checking whether 풜 possesses nontrivial invariant subspaces and, consequently, coincides with the full algebra by Burnside's theorem. This paper extends the class to the largest subclass of Jordan matrices on which the algorithm works correctly. Examples demonstrating different types of behavior of other matrix systems are provided. [ABSTRACT FROM AUTHOR]

In the paper, we consider a one-dimensional difference Schrödinger equation ψ(z+h)+ψ(z−h)+ λυ(z)ψ(z) = Eψ(z) with a periodic potential υ. In the case where the potential in a neighborhood of R is real analytic, as well as in the case where, the potential has a finite number of simple poles per period, for small values of the coupling constant λ, we describe the asymptotics of a monodromy matrix. [ABSTRACT FROM AUTHOR]

The paper provides a characterization of linear operators preserving column majorization of the (0, 1)-vectors. In addition, such operators are characterized explicitly in the case where they are given by special matrices, namely, by (±1)-matrices of order not exceeding 10 or (0,±1)-matrices of order not exceeding 5. Several related combinatorial matrix theory results also are proved. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of determining the number of subalgebras in the matrix algebra that are generated by nonderogatory matrices and are distinct up to similarity. A similarity criterion in terms of isomorphism of quotient algebras of the algebra of polynomials for matrix algebras of the type indicated is established. The existence of infinite fields for which the number of distinct algebras is infinite for all values of the matrix order is proved. For algebraically closed fields, the field of reals, and finite fields of sufficiently large cardinalities, the number of distinct algebras generated by nonderogatory matrices is determined as a function of the matrix order. [ABSTRACT FROM AUTHOR]

The paper suggests and illustrates a simple unified approach to deriving upper bounds for the dominant eigenvalues of the so-called PF matrices (or matrices with the Perron–Frobenius property) from those for the Perron root of a nonnegative matrix. [ABSTRACT FROM AUTHOR]

The paper interrelates the simultaneous triangularization problem for matrix pairs with the Paz problem and known results on the length of the matrix algebra. The length function is applied to the Al'pin–Koreshkov algorithm, and it is demonstrated how its multiplicative complexity can be reduced. An asymptotically superior procedure for verifying the simultaneous triangularizability of a pair of complex matrices is provided. The procedure is based on results on the lengths of upper triangular matrix algebras. Also the definition of the hereditary length of an algebra is introduced, and the problem of computing the hereditary lengths of matrix algebras is discussed. [ABSTRACT FROM AUTHOR]

The paper continues the study of the recently introduced class of SDD1 matrices. The class of general SDD1 matrices and its three subclasses are considered. In particular, it is shown that SDD1 matrices are nonsingular ℌ-matrices. Also parameter-free upper bounds for the l∞-norm of the inverses to SDD1 matrices are derived. The block triangular form to which any SDD1 matrix can be brought by a symmetric permutation of its rows and columns is described. [ABSTRACT FROM AUTHOR]

This paper starts a series of publications devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices. The paper considers rank factorizations and, in particular, the relatively irreducible and ΔW-2 factorizations, which are used in solving spectral problems for two-parameter polynomial matrices F(λ, µ). Algorithms for computing these factorizations are suggested and applied to computing points of the regular, singular, and regular-singular spectra and the corresponding spectral vectors of F(λ, µ). The computation of spectrum points reduces to solving algebraic equations in one variable. A new method for computing spectral vectors for given spectrum points is suggested. Algorithms for computing critical points and for constructing a relatively free basis of the right null-space of F(λ, µ) are presented. Conditions sufficient for the existence of a free basis are established, and algorithms for checking them are provided. An algorithm for computing the zero-dimensional solutions of a system of nonlinear algebraic equations in two variables is presented. The spectral properties of the ΔW-2 method are studied. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

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 Velázquez, 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. [ABSTRACT FROM AUTHOR]

The paper introduces new classes of nonsingular matrices, which contain some known subclasses of the class of nonsingular H-matrices, such as the Nekrasov, Q-Nekrasov, {P1, P2}-Nekrasov, and DZ matrices. For matrices in the classes introduced, upper bounds for ‖A−1‖∞ are derived (in a unified manner) and shown to improve the known bounds for matrices from the corresponding subclasses of the nonsingular H-matrices. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

The paper considers the so-called P-Nekrasov and {P1, P2}-Nekrasov matrices, defined in terms of permutation matrices P, P1, P2, which generalize the well-known notion of Nekrasov matrices. For such matrices A, available upper bounds on ‖A−1‖∞ are recalled, and new upper bounds for the P-Nekrasov and {P1, P2}-Nekrasov matrices are suggested. It is shown that the latter bound generally improves the earlier bounds, as well as the bound for the inverse of a P-Nekrasov matrix and the classical bound for the inverse of a strictly diagonally dominant matrix. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The paper investigates linear functionals ϕ : ℝn → ℝ preserving a set ℳ⊆ ℝ, i.e., ϕ : ℝn → ℝ such that ϕ(v) ∈ ℳ for any vector u ∈ ℝn with components in ℳ. For various types of subsets of real numbers, characterizations of the linear functionals that preserve them are obtained. In particular, the sets ℤ,ℚ, ℤ+,ℚ+,ℝ+, several infinite sets of integers, bounded and unbounded intervals, and finite subsets of real numbers are considered. A characterization of linear functionals preserving a set ℳ also allows one to describe the linear operators preserving matrices with entries from this set, i.e., the operators Φ : Mn,m → Mn,m such that all entries of the matrix Φ(A) belong to ℳ for any matrix A ∈ Mn,m with all entries in ℳ. As an example, linear operators preserving (0, 1)-, (±1)-, and (±1, 0)-matrices are characterized. [ABSTRACT FROM AUTHOR]

The paper considers the problem of computing the lengths of matrix incidence algebras over a field whose cardinality is strictly less than the matrix order n. For n = 3, 4, the lengths of all such algebras over the field of two elements are determined. In the case where the ground field and the number n are arbitrary but the Jacobson radical of the algebra has nilpotency index 2, an upper bound for the length is provided. In addition, the incidence algebras isomorphic to a direct sum of triangular matrix algebras of order 2 and an algebra of diagonal matrices are considered. It is shown that the lengths of these algebras over the field of two elements can take only two distinct values, which can be determined exactly. Moreover, the diagonal number of a matrix incidence algebra is introduced and bounded above. [ABSTRACT FROM AUTHOR]

The paper continues the study of block generalizations of Nekrasov matrices and introduces two new classes of the so-called G ˜ N and BJN matrices and compares them with the previously introduced class of GN matrices. Different properties of G ˜ N and BJN matrices are established. In particular, it is proved that the classes G ˜ N and {BJN} are closed with respect to Schur complements and monotone with respect to block partitioning. Also upper bounds for the norms ‖A−1‖∞ of inverses of GN, G ˜ N , and BJN matrices A are considered. General results obtained are specialized to the case of block two-by-two matrices with scalar first diagonal block. [ABSTRACT FROM AUTHOR]

The paper discusses possible forms of square matrices whose rows are pairwise orthogonal and the same is true of their columns. This discussion is applied to the problem of finding conditions under which a nonsingular binormal matrix is unitoid. A square matrix A is said to be binormal if the matrices AA∗ and A∗A commute. A square matrix is said to be unitoid if it can be brought to diagonal form by a (Hermitian) congruence. [ABSTRACT FROM AUTHOR]

The paper computes the Hilbert basis of the cone constructed from matrices describing generic situations, i.e., such vector subspaces in a finite direct sum of finite-dimensional subspaces that are in generic position with respect to the direct summands. [ABSTRACT FROM AUTHOR]

The paper presents new nonsingularity conditions for n ×n matrices, which involve a subset S of the index set {1,..., n} and take into consideration the matrix sparsity pattern. It is shown that the matrices satisfying these conditions form a subclass of the class of nonsingular H -matrices, which contains some known matrix classes such as the class of doubly strictly diagonally dominant (DSDD) matrices and the class of Dashnic–Zusmanovich type (DZT) matrices. The nonsingularity conditions established are used to obtain the corresponding eigenvalue inclusion sets, which, in their turn, are used in deriving new inclusion sets for the singular values of a square matrix, improving some recently suggested ones. [ABSTRACT FROM AUTHOR]

The paper is mainly devoted to studying the so-called Dashnic–Zusmanovich type (DZT) matrices, introduced recently. Interrelations among the DZT matrices and related subclasses of the class of nonsingular H -matrices, namely, the Dashnic–Zusmanovich (DZ) and S-SDD matrices are considered. Upper bounds for the l∞-norms of the inverses to DZT, DZ, and strictly diagonally dominant (SDD) matrices are obtained. A new eigenvalue inclusion set is provided. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The paper analyzes the product of m regular permutation groups G1·... · Gm, where m ≥ 2 is a natural number. Each of the regular permutation groups is a subgroup of the symmetric permutation group S(Ω) of degree |Ω| for the set Ω. M. M. Glukhov proved that for k = 2 and m = 2, 2-transitivity of the product G1· G2 is equivalent to the absence of zeros in the corresponding square matrix with the number of rows and columns equal to |Ω| − 1. Also M. M. Glukhov has given necessary conditions of 2-transitivity of such a product of regular permutation groups.In this paper, we consider the general case for any natural m and k such that m ≥ 2 and k ≥ 2. It is proved that k-transitivity of the product of regular permutation groups G1·... · Gm is equivalent to the absence of zeros in the square matrix with the number of rows and columns equal to (|Ω| − 1)!/(|Ω| − k)!. We obtain correlation between the number of arcs corresponding to this matrix and a natural number l such that the product (PsQt)l is 2-transitive, where P,Q ⊆ S(Ω) are some regular permutation groups and the permutation st is an (|Ω| − 1)-cycle. We provide an example of the building of AES ciphers such that their round transformations are k-transitive on a number of rounds. [ABSTRACT FROM AUTHOR]

We study the problem of realization of a given generalized oscillator by a system of N generalized oscillators of a different type. We consider a generalized oscillator related to a fixed system of orthogonal polynomials that are determined by three-term recurrent relations and the corresponding three-diagonal Jacobi matrix J. The case N =2 was considered in a previous paper. It was shown that in this case the orthogonality measure is symmetric with respect to rotation at angle π. In this paper, we consider the case N =3. We prove that such a problem has a solution only in two cases. In the first case, the Jacobi matrix related to the given “composite” generalized oscillator has block-diagonal form and consists of similar 3×3 blocks. In the second (more interesting) possible case, the Jacobi matrix is not block-diagonal. For this matrix, we construct the corresponding system of orthogonal polynomials. This system decomposes into three series which are related to Chebyshev polynomials of the second kind. The main result of the paper is a solution of the moment problem for the corresponding Jacobi matrix. In this case, the constructed measure is symmetric with respect to rotation at angle 2π/3. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

The paper continues the series of papers devoted to surveying and developing methods for solving problems for two-parameter polynomial and rational matrices. Different types of factorizations of two-parameter rational matrices (including irreducible and minimal ones), methods for computing them, and their applications to solving spectral problems are considered. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

Conditions for the solvability of the linear boundary-value problem for systems of differential-algebraic equations with the variable rank of the leading-coefficient matrix and the corresponding solution construction procedure have been found. [ABSTRACT FROM AUTHOR]

A square complex matrix A is said to be binormal if the associated matrices A*A and AA* commute. This matrix class yields a meaningful finite-dimensional extension of the concept of normality. The paper can be regarded as a survey of properties of binormal matrices. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 픻, the orthogonality graph of the ring Mn(픻) of n × n matrices over a skew field 픻 is connected and has diameter 4. If n = 2, then the graph of the ring Mn(픻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained. [ABSTRACT FROM AUTHOR]

The paper investigates the divisibility of the permanent function of (1,−1)-matrices by different powers of 2. It is shown that the Kräuter-Seifter bound is the best possible one for generic(1,−1)-matrices. [ABSTRACT FROM AUTHOR]

The paper presents a matrix representation of filter banks corresponding to spline wavelets with shifted supports. The matrix form of the decomposition and reconstruction filters simplifies the representation of nonuniform nonstationary wavelet transforms built on nonuniform grids on a finite interval. Such a representation of filters is used, for example, in constructing error-correcting codes. [ABSTRACT FROM AUTHOR]

The paper introduces the so-called generalized Nekrasov (GN) matrices, which provide a block extension of the ordinary Nekrasov matrices. Basic properties of GN matrices are studied. In particular, it is proved that the GN matrices form a subclass of nonsingular H-matrices and this subclass is closed with respect to Schur complements obtained by eliminating leading principal block submatrices. Also an upper bound for the l∞-norm of the inverse to a GN matrix, which generalizes the known bound for Nekrasov matrices, is obtained. The case of block two-by-two GN matrices with scalar first diagonal blocks, which prove to be Dashnic–Zusmanovich matrices of the first type, is considered separately. The bounds obtained are applied to SDD matrices. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

The paper considers some classes of block matrices that can be regarded as block generalizations of the class of $$ \mathrm{\mathscr{H}} $$ -matrices and interrelations among them. New bounds for the infinity norm of inverses for matrices in the classes under consideration are suggested. [ABSTRACT FROM AUTHOR]

The problem of finding (−1, 1)-matrices with vanishing permanent was posed by Edward Wang in 1974. This paper states and proves bounds on the number of negative entries in a matrix with zero permanent and minimal number of negative entries among all matrices of the same equivalence class. Then representatives of every equivalence class of matrices with zero permanent are found for n ≤ 5. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that for λ > 1, an additive map that strongly preserves the set of λ-scrambling matrices over the Boolean semiring B is a bijection. The general form of such a map over any antinegative commutative semiring with identity and without zero divisors is characterized. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

The paper considers some subclasses of the class of nonsingular H-matrices whose definitions involve matrix sparsity pattern. For matrices A in these subclasses, upper bounds for ‖A−1‖∞ are derived and shown to be sharper than the corresponding bounds ignoring matrix sparsity. Bibliography: 26 titles. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Double occurrence words play an important part in genetics in describing epigenetic genome rearrangements. A useful geometric representation for double occurrence words is provided by the so-called assembly graphs. The paper investigates properties of the incidence matrices that correspond to the assembly graphs. An explicit matrix characterization of the simple assembly graphs of a given structure and a series of constructions, using these graphs and important for genetic investigations, are provided. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The paper suggests upper bounds for the l norm of the inverses to block matrices belonging to certain subclasses of the class of block $$ \mathrm{\mathscr{H}} $$-matrices, which improve and supplement known results. [ABSTRACT FROM AUTHOR]

Let Qn(ℂ) denote the space of all skew-symmetric n × n matrices over the complex field ℂ. The paper characterizes the linear mappings T : Qn(ℂ) → Qn(ℂ) that satisfy the condition per(T(A)) = per(A) for all matrices A ∈ Qn(ℂ) and an arbitrary n > 4. [ABSTRACT FROM AUTHOR]

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of "almost" diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained. [ABSTRACT FROM AUTHOR]

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 ∀ A ∈ T n → U − 1 AU ∈ T n and also the matrices U ∈ GLn(ℂ) such that ∀ A ∈ T n → U − 1 A U ¯ ∈ T n. [ABSTRACT FROM AUTHOR]

This paper is a review of recent results for large random matrices with heavy-tailed entries. First, we outline the development of and some classical results in random matrix theory. We focus on large sample covariance matrices, their limiting spectral distributions, and the asymptotic behavior of their largest and smallest eigenvalues and their eigenvectors. The limits significantly depend on the finite or infiniteness of the fourth moment of the entries of the random matrix. We compare the results for these two regimes which give rise to completely different asymptotic theories. Finally, the limits of the extreme eigenvalues of sample correlation matrices are examined. [ABSTRACT FROM AUTHOR]

This paper presents a simple way for diagonalization of matrices in an arbitrary field, with which one can calculate functions of matrices. [ABSTRACT FROM AUTHOR]

It is known that the Nekrasov and S-Nekrasov matrices form subclasses of (nonsingular) H-matrices. The paper presents some necessary and sufficient conditions for a square matrix with complex entries to be a Nekrasov and an S-Nekrasov matrix. In particular, characterizations of the Nekrasov and S-Nekrasov matrices in terms of the diagonal column scaling matrices transforming them into strictly diagonally dominant matrices are obtained. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

The paper considers upper bounds for the infinity norm of the inverse for matrices in two subclasses of the class of (nonsingular) H-matrices, both of which contain the class of Nekrasov matrices. The first one has been introduced recently and consists of the so-called S-Nekrasov matrices. For S-Nekrasov matrices, the known bounds are improved. The second subclass consists of the socalled QN- (quasi-Nekrasov) matrices, which are defined in the present paper. For QN-matrices, an upper bound on the infinity norm of the inverses is established. It is shown that in application to Nekrasov matrices the new bounds are generally better than the known ones. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

Let TH be the space of ( T + H)-matrices of order n. The paper considers the following question: Which unitary matrices U satisfy the condition ∀A ∈ TH → U* AU ∈ TH? A criterion for verifying whether a given matrix U has this property is proposed. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

The paper establishes the invariance of the simplex-module algorithm for expanding real numbers α = (α1, …, αd) in multidimensional continued fractions under linear-fractional transformations α′=α1′…αd1=Uα with matrices U from the unimodular group GLd+1(ℤ). It is shown that the convergents of the transformed collections of numbers α′ satisfy the same recurrence relation and have the same approximation order. [ABSTRACT FROM AUTHOR]