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In this paper, we first investigate the matrix equation in a new fashion, where A, B, C, D and G are arbitrary matrices. Then, we establish some necessary and sufficient conditions for the existence of a solution of , where A, B are arbitrary matrices and C is a Hermitian matrix. In the special case when , we determine the general solution of , where A is an arbitrary matrix and C is a Hermitian matrix.

We consider matrices over the (nonnegative) integers and the integers mod k, and representations of them as linear combinations of permutation matrices. In both instances we describe and compare se...

To generalize D-nilpotent matrices that play a role in study of Druzkowski maps, we introduce quasi-D-nilpotent matrices. A matrix A is called quasi-D-nilpotent if there exists a subspace V of diagonal matrices of codimension 1 such that DA is nilpotent for all . It is proved that a quasi-D-nilpotent matrix has few nonzero principal minors. We also determine irreducible quasi-D-nilpotent matrices and the Frobenius normal forms of quasi-D-nilpotent matrices with respect to permutation similarity.

In this article, we present a sufficient condition for the existence of a unique positive definite solution of the non-linear matrix equation , where , (the set of all Hermitian positive definite matrices), are non-singular matrices and are order-preserving mappings. We give an example of a non-linear matrix equation of the above form (, A is a non-singular matrix, F is an order-preserving mapping), which is not solvable by previously known methods but solvable by using our new results.

This paper investigates the problem of establishing exact formulas for calculating the maximum and minimum ranks of matrix expressions or matrix equalities that involve two matrices and their generalized inverses of matrices. For a pair of matrices A and B such that the product AB is defined, two fundamental matrix expressions or matrix equalities composed by their generalized inverses are given by and , where and are the -inverses of A and B, respectively. Recently, the present author established a group of exact formulas for calculating the max–min ranks of the products , and derived many fundamental algebraic properties of from the rank formulas. As a continuation, we study in this paper the ranks of the multiple matrix products , , , , , as well as the matrix differences , , , . All these kinds of problems are in fact to characterize algebraic properties of certain matrix-valued functions and we need to utilize a variety of known matrix rank formulas during this tedious work.

In this paper, we show some necessary and/or sufficient conditions so that AB and/or BA are core matrices, whenever A and B are core complex matrices (a matrix A is a core matrix, that is a matrix of index one, if ). More specifically, we also analyse some similar situations where A, B, AB and/or BA are diagonalizable matrices or EP matrices. Moreover, we provide, as an application, two results concerning the reverse order law for the Moore–Penrose inverse of the product of two matrices.

The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms.

Let and denote the group of rth roots of unity, the set of all matrices over , the matrix with all entries 1 and the matrix with in the (1, 1)-entry and 1 elsewhere, respectively. Let . Wang proved that if and only if A can be obtained from by a finite sequence of elementary operations. In this paper, we generalize this result for matrices over by showing that the absolute value of permanent function achieves the maximum value, n!, if and only if A is the matrix obtained from by the certain elementary operations. Also, we prove that if , then and . Among other results we prove that for a complex Hadamard matrix .

We investigate column finite matrices over fields of characteristic different from 2. We prove that every such matrix can be written as a sum of at most 10 square-zero matrices.

Let be a symmetric matrix. Stochastic estimates have been introduced in the literature for estimating the trace of a function of the matrix A by computing the mean value of the quantities xTif(A)xi, where xi are appropriate sample vectors forming the columns of a sample matrix X. In this work, the selection of the vectors xi from the columns of a Hadamard matrix is discussed and the employment of the Hadamard matrices for the construction of the sample matrix X is studied and tested throughout several experiments. In addition, the combination of Hadamard matrices with a central-composite design (CCD) for the construction of the sample matrix X is introduced.

Let denote the set of matrices with entries in . We write to denote the subset of , all of whose entries are non-negative. Let denote the set of all real symmetric matrices. A matrix is said to be completely positive if there is some matrix such that . The CP-rank of the matrix is the smallest such that for some . In this article, we shall investigate the linear operators on that preserve sets of matrices defined by the CP-rank. We classify those that preserve the CP-rank function, those that preserve the set of CP-rank-1 matrices, those that preserve the sets of CP-rank-1 matrices and the set of CP-rank-2 matrices, and those that strongly preserve the set of CP-rank-1 matrices.

A state feedback is proposed to analyse the stability of a matrix polynomial in closed loop. First, it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable. Following the ideas of the Routh–Hurwitz stability procedure for scalar polynomials, certain continued-fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the Euclid’s division algorithm which leads to an extension of the well-known Routh–Hurwitz stability criteria but this time in terms of matrix coefficients. After that, stability of the closed-loop matrix polynomial is guaranteed based on a Corollary of a Lyapunov Theorem. The sufficient stability conditions are: (i) The matrices of one column of the presented array must be symmetric and positive definite and (ii) the matrices of the cascade realisation must satisfy a commutative condition. These stability conditions are also necessary for matrix polynomial of second order...

In this paper, the linear matrix equation is considered, where , are given large matrices and , are right-hand side and unknown matrices, respectively. The global least squares algorithm is applied to approximate the solution of this group of matrix equations. The right–left preconditioned global least squares algorithm is presented for obtaining the approximate solution of the mentioned matrix equation. This preconditioner is based on the C-orthogonalization process, where C is a symmetric positive definite matrix. Also, the iterative method and its preconditioned algorithm are proposed to find the approximate generalized inverses of nearly singular matrices and rectangular matrices. Finally, some numerical experiments are given to illustrate the efficiency of the new preconditioners.

This paper focuses on L-structured quaternion matrices. L-structured real matrices, conditions for the existence of solutions and the general solution of linear matrix equations were studied in the paper [Magnus JR. L-structured matrices and linear matrix equations, Linear Multilinear Algebra 1983;14:67–88]. In this paper, we present a theoretical study extending L-structured real matrices to L-structured quaternion matrices, and introduce some L-structured quaternion matrices. Based on them, we then discuss their applications in quaternion matrix equations.

Let be a non-negative matrix. Denote by , the infimum of those satisfying the following inequality:where , and is the Fibonacci numbers sequence and is a decreasing, non-negative sequence of real numbers. In this paper, first, the Fibonacci weighted sequence space is introduced. Then, we focus on the evaluation of , where is the Hausdorff matrix operator or the Norlund matrix operator or the transpose of the Norlund matrix operator. For the case of Hausdorff matrices, a Hardy-type formula is established as an upper estimate. Also, a general upper estimate is established for the case of Norlund matrices and their transpose. In particular, we apply our results to Cesaro matrices, Holder matrices, Euler matrices and Gamma matrices.

The Wedderburn rank reduction formula is a powerful method for developing matrix factorizations and many fundamental numerical linear algebra processes. We present a new interpretation of the Wedderburn rank reduction formula and its associated biconjugation process and see a more extensive result of the formula. In doing this, we propose a new formulation based on the null space transformations on rows and columns of A simultaneously, and show several matrix factorizations that can be derived from the Wedderburn rank reduction formula. We also present a generalization of the biconjugation process and compute banded and Hessenberg factorizations. Using the new formulation, we compute the WZ and ZW factorizations of a nonsingular matrix as well as the and the factorizations of a symmetric positive definite matrix. Then, we develop the integer Wedderburn rank reduction formula and its integer biconjugation process and present a class of algorithms for computing all integer matrix factorizations such as the integer LU factorization and the Smith normal form of an arbitrary integer matrix.

The linear algebraic theory of the Pascal and Vandermonde matrix is well developed by many authors. In the last two decades many interrelations have been discovered between the mentioned matrices, their generalizations and the Stirling matrices. We follow this direction and discover new matricial relations by using the so-called r-Whitney numbers. Along this way, we develop two natural extensions of the Vandermonde matrix with which we can study and evaluate successive power sums of arithmetic progressions and win new identities for the r-Whitney numbers.

We introduce nearly symmetric-decomposable matrices, which are a symmetric analogue of nearly decomposable matrices. The defining matrix property is related to 2-matching covered and minimally 2-matching covered properties of a graph (possibly containing loops). We develop constructions and properties of nearly symmetric-decomposable -matrices, some of which relate to the location and number of nonzero entries on the main diagonal.

The mapping $\Phi_n(A,B)=AB-BA$, where the matrices $A,B \in \mathbb{C}^{2n \times 2n}$ are skew-Hamiltonian with respect to transposition, is studied. Let $\mathcal{C}_n$ be the range of $\Phi_n$: we give an implicit characterization of $\mathcal{C}_n$, obtaining results that find an application in algebraic geometry. Namely, they are used in [R. Abuaf and A. Boralevi, Orthogonal bundles and skew-Hamiltonian matrices, In Preparation] to study orthogonal vector bundles. We also give alternative and more explicit characterizations of $\mathcal{C}_n$ for $n \leq 3$. Moreover, we prove that for $n \geq 4$ the complement of $\mathcal{C}_n$ is nowhere dense in the set of $2n$-dimensional Hamiltonian matrices, denoted by $\mathcal{H}_n$, implying that almost all matrices in $\mathcal{H}_n$ are in $\mathcal{C}_n$ for $n \geq 4$. Finally, we show that $\Phi_n$ is never surjective as a mapping from $\mathcal{W}_n \times \mathcal{W}_n$ to $\mathcal{H}_n$, where $\mathcal{W}_n$ is the set of $2n$-dimensional skew-Hamiltonian matrices. Along the way, we discuss the connections of this problem with several existing results in matrix theory.

For -matrices in general, and for -matrices with a constant number of 1s in each row and column, we determine the possible sets of traces of such matrices as its rows are permuted. Except for some special matrices, which are identified, the set of such traces is a consecutive set of integers from .

We investigate combinatorial matrix problems that are related to restricted integer partitions. They arise from Survo puzzles, where the basic task is to fill an m×n table by integers 1, 2, … , mn, so that each number appears only once, when the column sums and the row sums are fixed. We present a new computational method for solving Survo puzzles with binary matrices that are recoded and combined using the Hadamard, Kronecker, and Khatri–Rao products. The idea of our method is based on using the matrix interpreter and other data analytic tools of Survo R, which represents the newest generation of the Survo computing environment, recently implemented as a multiplatform, open source R package. We illustrate our method with detailed examples.

This paper deals with the modern development of matrices, linear transformations, quadratic forms and their applications to geometry and mechanics, eigenvalues, eigenvectors and characteristic equations with applications. Included are the representations of real and complex numbers, and quaternions by matrices, and isomorphism in order to show that matrices form a ring in abstract algebra. Some special matrices, including Hilbert’s matrix, Toeplitz’s matrix, Pauli’s and Dirac’s matrices in quantum mechanics, and Einstein’s Pythagorean formula are discussed to illustrate diverse applications of matrix algebra. Included also is a modern piece of information that puts mathematics, science and mathematics education professionals at the forefront of advanced study and research on linear algebra and its applications.

For any pattern of the specified entries, an explicit finite list of polynomial inequalities in the specified entries is given that characterizes the completability of a partial positive matrix to a TP matrix. Each polynomial happens to be a monomial, and an algorithm, whose input is the pattern, is given for finding the polynomials. The method uses some new partial orders on matrices and the logarithmic method to reduce the question to determining the generators of a certain finitely generated, pointed cone.

Recently, a new class of matrices, called mixed matrices, that unifies the Z-matrices and symmetric matrices has been identified. They share the property that when the leading principal minors are positive, all principal minors are positive. It is natural to ask what other properties of M-matrices and positive definite matrices are enjoyed by mixed matrices as well. Here, we show that mixed P-matrices satisfy a broad family of determinantal inequalities, the Koteljanskii inequalities, previously known for those two classes. In the process, other properties of mixed matrices are developed, and consequences of the Koteljanskii inequalities are given.

In this note, we present a generalization of some results concerning the spectral properties of a certain class of block matrices. As applications, we study some of its implications on nonnegative matrices, doubly stochastic matrices and graph theory namely on graph spectra and graph energy.

This paper studies the perturbation theory for spectral projections of Hermitian matrix pairs $(H, M)$, where $H$ is non-singular Hermitian matrix which can be factorized as $H = G J G^*$, $J = \diag(\pm 1)$, and $M$ is positive definite. The class of allowed perturbations is so restricted that the corresponding perturbed pair $(\wtd H, \wtd M)= (H+\delta H, M+ \delta M)$ must have the form $\wtd H = \wtd G J \wtd G^*$, $J = \diag(\pm 1)$ and $\wtd M$ is positive definite. The main contribution of the paper is a $\sin\Theta$ theorem which generalizes the main result from the first part of the paper to this more general setting. Our estimate, in its most general form, depends on a uniform norm bound on a set of all $J$-unitary matrices which diagonalize $G^*G$. The second main contribution is a new sharp uniform estimate of a norm of a all $J$-unitary matrix which diagonalize $G^*G$ such that $H=G^*JG$ is a {; ; ; ; ; ; \it{; ; ; ; ; ; quasi-definite}; ; ; ; ; ; }; ; ; ; ; ; matrix. The case of a quasi-definite pair is therefore the case where our bounds are most competitive. We present numerical experiments to corroborate the theory.

In this article, the invertible matrices over commutative semirings are studied. Some properties and equivalent descriptions of the invertible matrices are given and the inverse matrix of an invertible matrix is presented by analogues of the classic adjoint matrix. Also, Cramer's rule over a commutative semiring is established. The main results obtained in this article generalize the corresponding results for matrices over commutative rings, for lattice matrices, for incline matrices, for matrices over zerosumfree semirings and for matrices over additively regular semirings.

For a positive integer , let be the set of complex matrices. Suppose are positive integers and . Denote by the -numerical range of a matrix . It is shown that a linear map satisfiesfor all and if and only if there is a unitary such that one of the following holds. For all , , and for all , , where (1) is the identity map or the transposition map , or (2) and has the form or .

In this paper, we present a simple method to construct examples of infinite matrices belonging to different classes of matrix spaces. Moreover, we introduce a scale of matrix spaces which extends the well-known scale of classical Lebesgue spaces.

We formulate several open problems and conjectures involving acyclic doubly stochastic matrices with minimum permanent and the minimum second largest eigenvalue. New characterizations and illustrative examples of these permanental minimizing matrices are provided in this article. In addition, we show that the adjacency matrix of any tree, with more than 2 vertices, plus the identity matrix is not cohesive.

In this article, we present a multi-start approach to solve a determinant maximization problem of an integer matrix variable that has a constant trace. In order to enforce that elements of the matrix variable are integer, we add a nonlinear repeller term to the objective function. To solve this nonlinear optimization problem we implement our sequential conic trust-region-based algorithm within a multi-start framework. To demonstrate the potential of our approach, we use a determinant maximization model from crystallography.

The spread of a matrix is defined as the maximum absolute value of the difference between any two eigenvalues of that matrix. Of interest, here is the spread of n × n normal matrices with entries restricted to a closed interval. In particular, we are interested in the classes of real symmetric matrices and real skew-symmetric matrices. For each of these classes, we investigate the maximum spread under certain conditions, such as small rank, and determine the structure of these matrices when their spread attains this maximum value.

Let $(P,\preceq)$ be a lattice, $S$ a finite subset of $P$ and $f_1,f_2,...,f_n$ complex-valued functions on $P$. We define row-adjusted meet and join matrices on $S$ by $(S)_{f_1,...,f_n}=(f_i(x_i\wedge x_j))$ and $[S]_{f_1,...,f_n}=(f_i(x_i\vee x_j))$. In this paper we determine the structure of the matrix $(S)_{f_1,...,f_n}$ in general case and in the case when the set $S$ is meet closed we give bounds for $\text{rank} (S)_{f_1,...,f_n}$ and present expressions for $\det (S)_{f_1,...,f_n}$ and $(S)_{f_1,...,f_n}^{-1}$. The same is carried out dually for row-adjusted join matrix of a join closed set $S$.

A partial answer is given to the question of which expansive integer matrix dilations in ℝ2 have wavelet sets that are finite unions of convex sets. New results are given supporting a conjecture that among matrices with determinant greater than 2, it is exactly matrices that have a power equal to a scalar that have such wavelet sets.

Given a positive definite (p.d.) matrix with real entries, it is possible to construct a p.d. intraclass matrix whose diagonal and off-diagonal elements are chosen as the averages of the diagonal elements and off-diagonal elements of the former matrix. Exploiting the very special structure of the latter matrix various interesting propositions are established. Statistical applications of such matrices are surveyed.

A sign pattern matrix M with zero trace is primitive non-powerful if for some positive integer k, M k = J #. The base l(M) of the primitive non-powerful matrix M is the smallest integer k. By considering the signed digraph S whose adjacent matrix is the primitive non-powerful matrix M, we will show that if l(M) = 2, the minimum number of non-zero entries of M is 5n − 8 or 5n − 7 depending on whether n is even or odd.

Hub-dominant matrices are natural extensions of hub matrices. In this article we study eigengaps of the Gram matrix associated with a hub-dominant matrix. A class of hub-dominant matrices is then constructed by using equiangular tight frames.

The generalised coupled Sylvester matrix equations [image omitted] [image omitted] with unknown matrices X and Y, have important applications in control and system theory. Also it is well known that the reflexive and anti-reflexive matrices have wide applications in many fields. In this article, we consider the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices. First we propose two new matrix equations equivalent to the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices, respectively. Then two new iterative algorithms are proposed for solving these matrix equations. The convergence analysis of the proposed iterative algorithms is derived. Finally, some numerical examples are presented to illustrate the theoretical results of this article.

A bordered matrix is a two-by-two partitioned matrix with its lower-right corner equal to a null matrix. In this article, we present eight partitioned matrices consisting of the Moore–Penrose inverses of submatrices in a bordered matrix, and give necessary and sufficient conditions for the eight partitioned matrices to be generalized inverses of the bordered matrix.

This article presents a plantwide control structure design using dynamic performance-based optimization. The controlled and manipulated variable sets were obtained heuristically. The main plantwide control structures were established via dynamic optimization and the rest established via heuristic rules. The optimization problem was formulated as dynamic mixed integer nonlinear programming with objective function of measuring the control performance (ITAE of CVs) and the cost of manipulated variables (ITAE of MVs). The integer matrix in the formulation represents the plantwide control structure explicitly. The Tennessee Eastman (TE) process was selected to illustrate the proposed design procedure. The obtained control structures were evaluated via dynamic simulation in the face of various disturbances and set-point changing. The results were then compared with the control structure proposed by Luyben et al. (1999).

The structures of some important types of matrices over the quaternion skew field are discussed, and corresponding decompositions are obtained by virtue of real orthogonal matrices and real anti-symmetric matrices. In particular, the normal forms are found for several classes of quaternionic matrices.

Let M be an integer matrix with absolute values of all its eigenvalues being greater than 1. We give a characterization of compactly supported M-refinable splines f and the conditions that the shifts of f form a Riesz basis.

Let A be an n by n real symmetric matrix, and let α be a non-empty subset of {1, …, n}. The principal submatrix of A obtained by removing rows and columns indexed by α is denoted by A(α). If the nullity of A(α) is |α| more than that of A, then α is called a P-set of A, and the indices in α are called P-vertices of A. This article concerns P-vertices and P-sets of non-singular acyclic matrices A when the graph T of A is a path or a star. It was shown in [C.R. Johnson and B.D. Sutton, Hermitian matrices, eigenvalue multiplicities, and eigenvector components, SIAM J. Matrix Anal. Appl. 26 (2004), pp. 390–399] that each singular acyclic matrix of order n has at most n − 2 P-vertices. In this article we show that this does not hold for non-singular acyclic matrices by constructing non-singular acyclic matrices, whose graphs are T, having n − 1 (or n) P-vertices (these matrices also achieve the maximum size of a P-set over non-singular acyclic matrices whose graphs are T). In addition, it is shown that if n is ...

In this note we consider the question under which conditions all entries of the matrix I − (I + X)−1 are nonnegative in case matrix X is a real positive definite matrix. Sufficient conditions are presented as well as some necessary conditions. One sufficient condition is that matrix X −1 is an inverse M-matrix. A class of matrices for which the inequality holds is presented.

A successive method of descending degree has been presented by Guo, Liu and Jia in 2002 to judge whether or not a matrix is a generalized diagonally dominant matrix. In this paper, an alternative approach and comments on one of their examples are provided. Numerical examples are given to illustrate the algorithm processing.

Synthetic division is viewed as a change of basis for polynomials written under the Newton form. Then, the transition matrices obtained from a sequence of changes of basis are used to factorize the inverse of a bidiagonal matrix or a block bidiagonal matrix.

The computation of the McMillan degree and structure at infinity of a transfer function model is considered for the family of early design models, referred to as Structured Transfer Function (STF) matrices. Such transfer functions have certain elements fixed to zero, some elements being constant and other elements expressing some identified dominant dynamics of the system. For the family of large dimension STF matrices the computation of the generic McMillan degree and structure at infinity are considered using genericity arguments which lead to optimization problems of integer matrices. A novel approach is introduced here that uses the notion of “irreducibility” of integer matrices, which is developed as the equivalent of irreducibility (properness) of polynomial matrices. This new notion provides the means for exploiting the structure of integer matrices and enables the termination of searching processes in a reduced number of steps, thus leading to an efficient new algorithm for the computation of the ...

We show that the only compatible lattice order on a matrix ring over the integers for which the identity matrix is positive is (up to isomorphism) the usual, entrywise, lattice order. We also find a condition that guarantees that the only compatible lattice order on a matrix ring over the integers is formed by multiplying the positive cone of the usual, entrywise, lattice order by a matrix with positive entries. Using this condition, we show that such orders are the only compatible ones in the two-by-two case.

(2007). The Probability that a Matrix of Integers Is Diagonalizable. The American Mathematical Monthly: Vol. 114, No. 6, pp. 491-499.

By calculating the permanents for all Hadamard matrices of orders up to and including 28 we answer a problem posed by E.T.H. Wang and a similar question asked by H. Perfect. Both questions are answered by the existence of Hadamard matrices of order 20 which do not seem to be simply related but nevertheless have the same permanent. For orders up to and including 20 we also settle several other existence questions involving permanents of (+1, −1)-matrices. Specifically, we establish the lowest positive value taken by the permanent in these cases and find matrices which have equal permanent and determinant when such a matrix exists. Our results address Conjectures 19 and 36 and Problems 5 and 7 in Minc's well known catalogue of unsolved problems on permanents. We also include a little-known proof that there exists a (+1, −1)-matrix A of order n such that per(A) = 0 if and only if n + 1 is not a power of 2.