P-matrix

In mathematics, a P-matrix is a complex square [matrix (mathematics)|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 \geq 0.

Contents

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:

If \{u_1,...,u_n\} are the eigenvalues of an n-dimensional P-matrix, then
|arg(u_i)| < \pi - \frac{\pi}{n}, i = 1,...,n
If \{u_1,...,u_n\}, u_i \neq 0, i = 1,...,n are the eigenvalues of an n-dimensional P_0-matrix, then
|arg(u_i)| \leq \pi - \frac{\pi}{n}, i = 1,...,n

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.[1]

If the Jacobian of a function is a P-matrix, then the function is injective on any rectangular region of \mathbb{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 \sigma(A) = -\sigma(-A), the eigenvalues of these matrices are bounded away from the positive real axis.

Notes

  1. ^ Csizmadia, Zsolt; Illés, Tibor (2006). "New criss-cross type algorithms for linear complementarity problems with sufficient matrices" (pdf). Optimization Methods and Software 21 (2): 247–266. doi:10.1080/10556780500095009. MR2195759. http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf. 

References