# P-matrix

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 $\geq$ 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:

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.

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. MR 2195759. Unknown parameter |eprint= ignored (help); Unknown parameter |url2= ignored (help)