P-matrix

In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of -matrices, which are the closure of the class of P-matrices, with every principal minor 0. By a theorem of Kellogg, the eigenvalues of P- and - matrices are bounded away from a wedge about the negative real axis as follows: If are the eigenvalues of an n-dimensional P-matrix, where , then If , , are the eigenvalues of an n-dimensional -matrix, then The class of nonsingular M-matrices is a subset of the class of P-matrices. More precisely, all matrices that are both P-matrices and Z-matrices are nonsingular M-matrices. The class of sufficient matrices is another generalization of P-matrices. The linear complementarity problem has a unique solution for every vector q if and only if M is a P-matrix. This implies that if M is a P-matrix, then M is a Q-matrix. If the Jacobian of a function is a P-matrix, then the function is injective on any rectangular region of . A related class of interest, particularly with reference to stability, is that of -matrices, sometimes also referred to as -matrices. A matrix A is a -matrix if and only if is a P-matrix (similarly for -matrices). Since , the eigenvalues of these matrices are bounded away from the positive real axis.