pmatrix

Pmatrix, in mathematical literature, refers to a real square matrix with the property that all principal minors are strictly positive. For an n×n matrix A, every determinant det(A[I,I]) taken from any nonempty index set I ⊆ {1,…,n} must be positive; the empty minor is usually taken as 1. This definition places Pmatrices within the broader study of matrix classes used in optimization and numerical analysis.

Key properties include that Pmatrices are always nonsingular, since det(A) = det(A[{1..n},{1..n}]) > 0. They relate to other matrix classes: every symmetric positive definite matrix is a Pmatrix, because positive definiteness implies positive determinants of all principal minors. Conversely, a Pmatrix need not be symmetric or positive definite, so the property does not by itself enforce positive eigenvalues in the general (non-symmetric) case. There are related classes such as P0-matrices, where principal minors are nonnegative rather than strictly positive.

In linear complementarity problems (LCP), Pmatrices play a central role. The LCP with a Pmatrix M and a vector q often has well-posed behavior: in many formulations, a Pmatrix guarantees the existence of a solution and, in fact, uniqueness for every q. This makes Pmatrices a key object of study in numerical methods for equilibrium and optimization problems.

Typical examples include the identity matrix and symmetric positive definite matrices, such as [[2, -1], [-1, 2]], which has positive determinants for all principal minors. The concept also connects to other matrix families; for instance, every nonsingular M-matrix is a Pmatrix, illustrating overlap with stability-focused matrix theory.