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Semipositivity of matrices and linear maps relative to proper cones

Chandrashekaran Arumugasamy(chandru1782***at***gmail.com)
Sachindranath Jayaraman(sachindranathj***at***gmail.com)
Vatsalkumar Mer(vnm232657***at***gmail.com)

Abstract: For a proper cone $K$ in a finite dimensional real Hilbert space $V$, a square matrix $A$ (a linear map $L$) is said to be $K$-semipositive if there exists $d \in K^\circ$, the interior of $K$, such that $Ad \in K^\circ$ ($Ld \in K^\circ$). Our aim in this manuscript is to characterize $K$-semipositivity of matrices / linear maps relative to a proper cone. Among several results obtained, we characterize $K$-semipositivity in terms of products of the form $YX^{-1}$ for $K$-positive matrices / linear maps ($A(K \setminus \{0\}) \subseteq K^\circ$) with $X$ invertible, characterize semipositivity of matrices relative to the $n$-dimensional Lorentz cone $\mathcal{L}^n_{+}$, characterize semipositivity of the following three linear maps relative to the cone $\mathcal{S}^n_{+}$: $X \mapsto AXB$ (denoted by $M_{A,B}$), $X \mapsto AXB + B^tXA^t$ (denoted by $L_{A,B}$), where $A, B \in M_n(\mathbb{R})$, and $X \mapsto X - AXA^t$ (denoted by $S_A$, known as the Stein transformation). We prove that $M_{A,B}$ is semipositive if and only if $B = \alpha A^t$ for some $\alpha > 0$, the map $L_{A,B}$ is semipositive if and only if $A(B^t)^{-1}$ is positive stable. A particular case of our result generalizes Lyapunov's theorem. Decompositions of the above maps (when they are semipositive) in the form $L_1L_2^{-1}$, where $L_1$ and $L_2$ are both positive and invertible (assuming $A$ is invertible in the case of $S_A$) are presented. We also investigate properties of the semipositive cone $\mathcal{K}_A$ of a matrix / linear map and partially answer a question on invariance of $\mathcal{K}_A$ under $A$.

Keywords: Positivity and semipositivity of linear maps, proper cones, positive definite matrices, positive stable matrices, semidefinite linear complementarity problems, Lyapunov and Stein transformations, semipositive cone

Category 1: Complementarity and Variational Inequalities

Category 2: Linear, Cone and Semidefinite Programming

Citation: Manuscript Details: February/2018, 32 pages. MSC 2010: 15B48, 90C33 Author 1: Department of Mathematics, School of Mathematics & Computer Sciences, Central University of Tiruvarur- 610101, Tamilnadu, India. Authors 2 & 3: School of Mathematics, IISER Thiruvananthapuram, Maruthamala P.O, Vithura, Trivandrum - 695551, Kerala, India.

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Entry Submitted: 02/20/2018
Entry Accepted: 02/21/2018
Entry Last Modified: 02/20/2018

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