In mathematics, a P -matrix is a complex square matrix with every principal minor > 0. A closely related class is that of P 0 -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 P 0 - matrices are bounded away from a wedge about the negative real axis as follows:
If
{ u 1 , . . . , u n } are the eigenvalues of an
n -dimensional
P -matrix, where
n > 1 , then
| a r g ( u i ) | < π − π n , i = 1 , . . . , n If
{ u 1 , . . . , u n } ,
u i ≠ 0 ,
i = 1 , . . . , n are the eigenvalues of an
n -dimensional
P 0 -matrix, then
| a r g ( u i ) | ≤ π − π n , i = 1 , . . . , n 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 L C P ( M , q ) has a unique solution for every vector q if and only if M is a P -matrix.
If the Jacobian of a function is a P -matrix, then the function is injective on any rectangular region of R n .
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 σ ( A ) = − σ ( − A ) , the eigenvalues of these matrices are bounded away from the positive real axis.
