Necessary and sufficient conditions are presented for the commutativity equalities $A^{*}{A}^D=A^D{A}^{*}$, $A^{†}{A}^D=A^D{A}^{†}$, $A^{†}A{A}^D=A^{D}A{A}^{†}$, $A{A}^D{A}^{*}=A^{*}{A}^{D}A$ and so on to hold by using rank equalities of matrices. Some related topics are also examined.

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications,...

Let A and B be M-matrices satisfying A ≤ B and J = [A,B] be the set of all matrices C such that A ≤ C ≤ B, where the order is component wise. It is rather well known that if A is an M-matrix and B is an invertible M-matrix and A ≤ B, then aA + bB is an invertible M-matrix for all a,b > 0. In this article, we present an elementary proof of a stronger version of this result and study corresponding results for certain other classes as well.

A pair of simple bivariate inverse series relations are used by embedding machinery to produce several double summation formulae on shifted factorials (or binomial coefficients), including the evaluation due to Blodgett-Gessel. Their q-analogues are established in the second section. Some generalized convolutions are presented through formal power series manipulation.

Cet article présente trois résultats distincts. Dans une première partie nous donnons l’asymptotique quand $N$ tend vers l’infini des coefficients des polynômes orthogonaux de degré $N$ associés au poids ${\varphi }_{\alpha }\left(\theta \right)={|1-e^{i\theta }|}^{2\alpha }{f}_1\left(e^{i\theta }\right)$, où $f_1$ est une fonction strictement positive suffisamment régulière et $\alpha \>\frac{1}{2},\alpha \in ℝ\setminus \mathbb{N}$. Nous en déduisons l’asymptotique des éléments de l’inverse de la matrice de Toeplitz $T_N\left({\varphi }_{\alpha }\right)$ au moyen d’un noyau intégral $G_{\alpha }.$ Nous prolongeons ensuite un résultat de A. Böttcher et H. Windom relatif à l’asymptotique de la valeur propre...

Ce travail est une étude théorique d’opérateurs de Toeplitz dont le symbole est une fonction matricielle régulière définie positive partout sur le tore à une dimension. Nous proposons d’abord une formule d’inversion exacte pour un opérateur de Toeplitz à symbole matriciel, démontrée au moyen d’un théorème établi en annexe et donnant la solution du problème de la prédiction relatif à un passé fini pour un processus stationnaire du second ordre. Nous établissons ensuite, à partir de cet inverse, un...

The estimation procedures in the multiepoch (and specially twoepoch) linear regression models with the nuisance parameters that were described in [2], Chapter 9, frequently need finding the inverse of a $3\times 3$ partitioned matrix. We use different kinds of such inversion in dependence on simplicity of the result, similarly as in well known Rohde formula for $2\times 2$ partitioned matrix. We will show some of these formulas, also methods how to get the other formulas, and then we applicate the formulas in estimation...

Some useful tools in modelling linear experiments with general multi-way classification of the random effects and some convenient forms of the covariance matrix and its inverse are presented. Moreover, the Sherman-Morrison-Woodbury formula is applied for inverting the covariance matrix in such experiments.

In this study various Jacobians of transformations of singular random matrices are found. An alternative proof of Uhlig's first conjecture (Uhlig (1994)) is proposed. Furthermore, we propose various extensions of this conjecture under different singularities. Finally, an application of the theory of singular distributions is discussed.

The Linear Discriminant Analysis (LDA) technique is an important and well-developed area of classification, and to date many linear (and also nonlinear) discrimination methods have been put forward. A complication in applying LDA to real data occurs when the number of features exceeds that of observations. In this case, the covariance estimates do not have full rank, and thus cannot be inverted. There are a number of ways to deal with this problem. In this paper, we propose improving LDA in this...

The set of all $m\times n$ Boolean matrices is denoted by ${𝕄}_{m,n}$. We call a matrix $A\in {𝕄}_{m,n}$ regular if there is a matrix $G\in {𝕄}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${𝕄}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $min\{m,n\}\le 2$, then all operators on ${𝕄}_{m,n}$ strongly preserve regular matrices, and if $min\{m,n\}\ge 3$, then an operator $T$ on ${𝕄}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T\left(X\right)=UXV$ for all $X\in {𝕄}_{m,n}$, or $m=n$ and $T\left(X\right)=U{X}^{T}V$ for all $X\in {𝕄}_n$.

We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.