Chandrashekaran Arumugasamy

arXiv2019

A characterization of nonnegativity relative to proper cones

Let $A$ be an $m \times n$ matrix with real entries. Given two proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, we say that $A$ is nonnegative if $A(K_1) \subseteq K_2$. $A$ is said to be semipositive if there exists a $x \in K_1^\circ$ such that $Ax \in K_2^\circ$. We prove that $A$ is nonnegative if and only if $A+B$ is semipositive for every semipositive matrix $B$. Applications of the above result are also brought out.

math.FA↗

arXiv2018

Linear maps on $M_n(\mathbb{R})$ preserving Schur stable matrices

An $n \times n$ matrix $A$ with real entries is said to be Schur stable if all the eigenvalues of $A$ are inside the open unit disc. We investigate the structure of linear maps on $M_n(\mathbb{R})$ that preserve the collection $\mathcal{S}$ of Schur stable matrices. We prove that if $L$ is a linear map such that $L(\mathcal{S}) \subseteq \mathcal{S}$, then $ρ(L)$ (the spectral radius of $L$) is at most $1$ and when $L(\mathcal{S}) = \mathcal{S}$, we have $ρ(L) = 1$. In the latter case, the map $L$ preserves the spectral radius function and using this, we characterize such maps on both $M_n(\mathbb{R})$ as well as on $\mathcal{S}^n$.

math.FA↗