On the Stability of P -Matrices

We establish two sufficient conditions for the stability of a P -matrix. First, we show that a P -matrix is positive stable if its skew-symmetric component is sufficiently smaller (in matrix norm) than its symmetric component. This result generalizes the fact that symmetric P -matrices are positive stable, and is analogous to a result by Carlson which shows that sign symmetric P -matrices are positive stable. Second, we show that a P matrix is positive stable if it is strictly row (column) square diagonally dominant for every order of minors. This result generalizes the fact that strictly row diagonally dominant P -matrices are stable. We compare our sufficient conditions with the sign symmetric condition and demonstrate that these conditions do not imply each other. Notations Given an n×n matrix A, we denote by Aij its element in i row and j column. We denote by ‖A‖2 the two-norm of A, that is maxi∈{1,2,..,n} √ λi where {λ1, .., λn} is the set of eigenvalues of A A. For subsets α and β of {1, 2.., n}, we denote by A(α|β) the sub-matrix of A with elements {Aij} where i ∈ α and j ∈ β. If |α| = |β|, then we call det(A(α|β)) the minor corresponding to index sets α and β and denote it by A(α, β). If |α| = k, we call A(α, α) a principal minor of A of order k. Let R denote the N dimensional Euclidean space, and R+ denote the nonnegative orthant. Given a complex number x ∈ C, we denote its polar representation by x = l(x)e , where l(x) ∈ R+ is the length of x and arg(x) ∈ (−π, π] is its argument.