On some matrix equalities for generalized inverses with applications

Necessary and sufficient conditions are derived for the matrix equality A^-=PN^-Q to hold, where A^- and N^- are generalized inverses of matrices. Some consequences and applications are also given. In particular, necessary and sufficient conditions are derived for the additive decompositions C^-=A^-+B^- and to hold.     

On a matrix decomposition of Hartwig and Spindelböck

The main task of the paper is to demonstrate that Corollary 6 in [R.E. Hartwig, K. Spindelböck, Matrices for which A^∗ and A^† commute, Linear and Multilinear Algebra 14 (1984) 241-256] provides a powerful tool to investigate square matrices with complex entries. This aim is achieved, on the one hand, by obtaining several original results involving square matrices, and, on the other hand, by reestablishing some of the facts already known in the literature, often in extended and/or generalized forms. The particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

Vector least-squares solutions for coupled singular matrix equations

The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian's work and Lev-Ari's results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case.     

Weighted least squares solutions to general coupled Sylvester matrix equations

This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.     

Canonical forms for mixed symmetric-skewsymmetric quaternion matrix pencils

Canonical forms are described for pairs of quaternionic matrices, or equivalently matrix pencils, where one matrix is symmetric and the other matrix is skewsymmetric, under strict equivalence and symmetry respecting congruence. The symmetry is understood in the sense of a fixed involutory antiautomorphism of the skew field of the real quaternions; the involutory antiautomorphism is assumed to be nonstandard, i.e., other than the quaternionic conjugation. Some applications are developed, such as canonical forms for quaternionic matrices under symmetry respecting congruence, and canonical forms for matrices that are skewsymmetric with respect to a nondegenerate symmetric or skewsymmetric quaternion valued inner product.     

Remarks on the sharp partial order and the ordering of squares of matrices

In this note we revisit the sharp partial order introduced by Mitra [S.K. Mitra, On group inverses and the sharp order, Linear Algebra Appl. 92 (1987) 17-37]. We recall some already known facts from certain matrix decompositions and derive new statements, relating our discussion to recent results in the literature concerned with partial orders between matrices and their squares.     

Linear complexity algorithm for semiseparable matrices

A new linear complexity algorithm for general nonsingular semiseparable matrices is presented. For symmetric matrices whose semiseparability rank equals to 1 this algorithm leads to an explicit formula for the inverse matrix.Supported in part by the NSF Grant DMS 9306357     

On schur complements in an ep matrix

It is established that under certain conditions a Schur complement in an EP matrix is as well an EP matrix. As an application a decomposition of a partitioned matrix into a sum of EP matrices is given.     

General solution of certain matrix equations arising in filter design applications

In this work we present the explicit expression of all rectangular Toeplitz matrices B,C which verify the equation BB^H+CC^H=aI for some a>0. This matrix equation arises in some signal processing problems. For instance, it appears when designing the even and odd components of paraunitary filters, which are widely used for signal compression and denoising purposes. We also point out the relationship between the above matrix equation and the polynomial Bézout equation |B(z)|²+|C(z)|²=a>0 for |z|=1. By exploiting this fact, our results also yield a constructive method for the parameterization of all solutions B(z),C(z). The main advantage of our approach is that B and C are built without need of spectral factorization. Besides these theoretical advances, in order to illustrate the effectiveness of our approach, some examples of paraunitary filters design are finally given.     

The centro-invertible matrix: A new type of matrix arising in pseudo-random number generation

This paper defines a new type of matrix (which will be called a centro-invertible matrix) with the property that the inverse can be found by simply rotating all the elements of the matrix through 180 degrees about the mid-point of the matrix. Centro-invertible matrices have been demonstrated in a previous paper to arise in the analysis of a particular algorithm used for the generation of uniformly-distributed pseudo-random numbers.An involutory matrix is one for which the square of the matrix is equal to the identity. It is shown that there is a one-to-one correspondence between the centro-invertible matrices and the involutory matrices. When working in modular arithmetic this result allows all possible k by k centro-invertible matrices with integer entries modulo M to be enumerated by drawing on existing theoretical results for involutory matrices.Consider the k by k matrices over the integers modulo M. If M takes any specified finite integer value greater than or equal to two then there are only a finite number of such matrices and it is valid to consider the likelihood of such a matrix arising by chance. It is possible to derive both exact expressions and order-of-magnitude estimates for the number of k by k centro-invertible matrices that exist over the integers modulo M. It is shown that order (√N) of the N=M^(k2) different k by k matrices modulo M are centro-invertible, so that the proportion of these matrices that are centro-invertible is order (1/√N).     

On a subspace metric based on matrix rank

The metric between subspaces M,N⊆C_n,1, defined by δ(M,N)=rk(P_M-P_N), where rk(·) denotes rank of a matrix argument and P_M and P_N are the orthogonal projectors onto the subspaces M and N, respectively, is investigated. Such a metric takes integer values only and is not induced by any vector norm. By exploiting partitioned representations of the projectors, several features of the metric δ(M,N) are identified. It turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces, such as distance (also called gap), Frobenius distance, direct distance, angle, or minimal angle.     

On commutativity of projectors

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P₁P₂ = P₂P₁, where P₁ and P₂ denote projectors (i.e., idempotent matrices). The first problem was considered by Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, A property of orthogonal projectors, Linear Algebra Appl. 354 (2002) 35-39], who have shown that if P₁ and P₂ are orthogonal projectors (i.e., Hermitian idempotent matrices), then in all nontrivial cases a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute. In the present paper a generalization of this result is proposed and validity of the equivalence between commutativity property and any equality involving two linear combinations of two any length products having orthogonal projectors P₁ and P₂ as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of the sum P₁ + P₂ and the difference P₁ − P₂ of (not necessary orthogonal) commuting projectors P₁ and P₂. The results obtained supplement those provided in Section 4 of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142].     

Orthogonal similarity of a real matrix and its transpose

Any square matrix over a field is similar to its transpose and any square complex matrix is similar to a symmetric complex matrix. We investigate the situation for real orthogonal (respectively complex unitary) similarity.     

How real is your matrix?

For scalars there is essentially just one way to define reality, real part and to measure nonreality. In this paper various ways of defining respective concepts for complex-entried matrices are considered. In connection with this, products of circulant and diagonal matrices often appear and algorithms to approximate additively and multiplicatively with them are devised. Multiplicative structures have applications, for instance, in diffractive optics, preconditioning and fast Fourier expansions.     

Determinant of the distance matrix of a tree with matrix weights

Let T be a tree with n vertices and let D be the distance matrix of T. According to a classical result due to Graham and Pollack, the determinant of D is a function of n, but does not depend on T. We allow the edges of T to carry weights, which are square matrices of a fixed order. The distance matrix D of T is then defined in a natural way. We obtain a formula for the determinant of D, which involves only the determinants of the sum and the product of the weight matrices.     

Nonnegative determinant of a rectangular matrix: Its definition and applications to multivariate analysis

It is well known that the determinant of a matrix can only be defined for a square matrix. In this paper, we propose a new definition of the determinant of a rectangular matrix and examine its properties. We apply these properties to squared canonical correlation coefficients, and to squared partial canonical correlation coefficients. The proposed definition of the determinant of a rectangular matrix allows an easy and straightforward decomposition of the likelihood ratio when given sets of variables are partitioned into row block matrices. The last section describes a general theorem on redundancies among variables measured in terms of the likelihood ratio of a partitioned matrix.     

An optimum iterative method for solving any linear system with a square matrix

A method is presented to solveAx=b by computing optimum iteration parameters for Richardson's method. It requires some information on the location of the eigenvalues ofA. The algorithm yields parameters well-suited for matrices for which Chebyshev parameters are not appropriate. It therefore supplements the Manteuffel algorithm, developed for the Chebyshev case. Numerical examples are described.