In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of P-matrices, with every principal minor \geq 0.

Spectra of P-matrices

By a theorem of Kellogg, the eigenvalues of P- and P_0- matrices are bounded away from a wedge about the negative real axis as follows:

Remarks

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 P^{(-)}-matrices, sometimes also referred to as N-P-matrices. A matrix A is a P^{(-)}-matrix if and only if (-A) is a P-matrix (similarly for P_0-matrices). Since, the eigenvalues of these matrices are bounded away from the positive real axis.