P-matrix

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:


 * If $$\{u_1,...,u_n\}$$ are the eigenvalues of an $n$-dimensional $P$-matrix, where $$n>1$$, 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.

The linear complementarity problem $$\mathrm{LCP}(M,q)$$ 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 $$\mathbb{R}^n$$.

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.