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In the setting of real square matrices, it is known that, if $A$ is a singular irreducible $M$-matrix, then the only nonnegative vector that belongs to the range space of $A$ is the zero vector. In this paper, we prove an analogue of this result for the Lyapunov and the Stein operators.

A real square matrix $A$ of order $n \times n~ (n \geq 3)$ is called an $F_0$-matrix, if it is a $Z$-matrix (off-diagonal entries nonpositive), all of whose principal submatrices of orders at most $n-2$ are $M$-matrices while there is at least one principal submatrix of order $n-1$, which is an $N_0$-matrix. An $M$-matrix is a $Z$-matrix with the property that the real parts of all its eigenvalues are nonnegative. An $N_0$-matrix, in turn, is characterized by the fact that it is an invertible $Z$-matrix whose inverse is (entrywise) nonpositive. The first aim of this article is to present a short survey of some subclasses of $Z$-matrices, pertinent to the second objective, where new results concerning $F_0$-matrices are presented., Comment: 25 pages

The aim of this note is to show that certain number theoretic inequalities due to Nesbitt and Shapiro have noncommutative counterparts involving positive definite matrices., Comment: To appear in Advances in Operator Theory. 10 pages, no figures

Let $H$ be a real Hilbert space. In this short note, using some of the properties of bounded linear operators with closed range defined on $H$, certain bounds for a specific convex subset of the solution set of infinite linear complementarity problems, are established.

A number $s$ is the sum of the entries of the inverse of an $n \times n, (n \geq 3)$ upper triangular matrix with entries from the set $\{0, 1\}$ if and only if $s$ is an integer lying between $2-F_{n-1}$ and $2+F_{n-1}$, where $F_n$ is the $n$th Fibonacci number. A generalization of the sufficient condition above to singular, group invertible matrices is presented.

A real square matrix $A$ is called a $Q$-matrix if the linear complementarity problem $LCP(A,q)$ has a solution for all $q \in \mathbb{R}^n$. This means that for every vector $q$ there exists a vector $x$ such that $x \geq 0, y=Ax+q\geq 0$ and $x^Ty=0$. A well known result of Karamardian states that if the problems $LCP(A,0)$ and $LCP(A,d)$ for some $d\in \mathbb{R}^n, d >0$ have only the zero solution, then $A$ is a $Q$-matrix. By relaxing the condition on $d$ and imposing a condition on the solution vector $x$ in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, requiring that these two modified problems have only zero as a solution. In this article, a systematic treatment of Karamardian matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of $Q$-matrices. A subclass of a recently introduced notion of $P_{\#}$-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of $P_{\#}$-matrices and make some fundamental contributions.

We address the question as to when it is true that $(A+B)^{\dagger}=A^{\dagger}+B^{\dagger},$ where $\dagger$ denotes the Moore-Penrose inverse. A similar question is addressed for the group inverse.

Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this short note, we study the structure of linear maps which preserve the set of all semipositive and minimal semipositive matrices.

This is a survey on certain results which bring about a connection between Fibonacci sequences on the one hand and the areas of matrix theory and quantum information theory, on the other., Comment: 35 pages

The problem of when the vanishing of a (generalized) Schur complement of a block matrix (corresponding to the leading principal subblock) implies that the other (generalized) Schur complement (corresponding to the trailing principal subblock) is zero, is revisited. A simple proof is presented. Absorption laws for two important classes of generalized inverses are considered next. Inheritance properties of the generalized Schur compements in relation to the absorption laws are derived. Inheritance by the generalized principal pivot transform is also studied., Comment: 25 pages

In this short note, we prove a formula for the group inverse of a block matrix and consider the pseudo principal pivot transform expressed in terms of group inverses. Extensions of the usual principal pivot transform, where the usual inverse is replaced by the Moore-Penrose inverse, were considered in the literature. The objective here is to derive the corresponding formulae for the group generalized inverse. These are expected to be useful in our future work.

In this short note, we prove some basic results on pseudo Schur complement and the pseudo principal pivot transform of a block matrix. Pseudo Schur complement and pseudo principal pivot ransform are extensions of the Schur complement and the principal pivot transform, respectively, where the usual inverse is replaced by the Moore-Penrose inverse. The objective is to record these results for use in future.