- Semipositivity of matrices and linear maps relative to proper cones Chandrashekaran Arumugasamy(chandru1782gmail.com) Sachindranath Jayaraman(sachindranathjgmail.com) Vatsalkumar Mer(vnm232657gmail.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. Download: [PDF]Entry Submitted: 02/20/2018Entry Accepted: 02/21/2018Entry Last Modified: 02/20/2018Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.