Generating and detecting matrices with positive principal minors
Michael J. Tsatsomeros · 2004
Abstract. A brief but concise review of methods to generate P-matrices (i.e., matrices having positive principal minors) is provided and motivated by open problems on P-matrices and the desire to develop and test efficient methods for the detection of P-matrices. Also discussed are operations that leave the class of P-matrices invariant. Some new results and extensions of results regarding P-matrices are included. Key words. P-matrix, positive definite matrix, M-matrix, MMA-matrix, H-matrix, B-matrix, totally positive matrix, mime, Schur complement, principal pivot transform, Cayley transform, linear complementarity problem, P-problem. AMS subject classifications. 15A48, 15A15, 15A57, 15-02, 90-08 1. Introduction. An n × n complex matrix A ∈Mn(C) is called a P-matrix if all its principal minors are positive. Recall that a principal minor is simply the determinant of a submatrix obtained from A when the same set of rows and columns are stricken out. The diagonal entries and the determinant of A are thus among its principal minors. We shall denote the class of complex P-matrices by P.