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 {\displaystyle P_{0}} -matrices, which are the closure of the class of P-matrices, with every principal minor ≥ ≥ --> {\displaystyle \geq } 0.
By a theorem of Kellogg,[1][2] the eigenvalues of P- and P 0 {\displaystyle P_{0}} - matrices are bounded away from a wedge about the negative real axis as follows:
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.[3]
The linear complementarity problem L C P ( M , q ) {\displaystyle \mathrm {LCP} (M,q)} has a unique solution for every vector q if and only if M is a P-matrix.[4] 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 R n {\displaystyle \mathbb {R} ^{n}} .[5]
A related class of interest, particularly with reference to stability, is that of P ( − − --> ) {\displaystyle P^{(-)}} -matrices, sometimes also referred to as N − − --> P {\displaystyle N-P} -matrices. A matrix A is a P ( − − --> ) {\displaystyle P^{(-)}} -matrix if and only if ( − − --> A ) {\displaystyle (-A)} is a P-matrix (similarly for P 0 {\displaystyle P_{0}} -matrices). Since σ σ --> ( A ) = − − --> σ σ --> ( − − --> A ) {\displaystyle \sigma (A)=-\sigma (-A)} , the eigenvalues of these matrices are bounded away from the positive real axis.
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