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P-matrix

Complex square matrix for which every principal minor is positive

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Source updated April 14, 2025

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.

01Spectra 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

$|\arg(u_{i})|\leq \pi -{\frac {\pi }{n}},\ i=1,...,n$

02Remarks

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.

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Sources and credits

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P-matrix

01Spectra of P-matrices

02Remarks

Sources and credits

Related topics

Routh-Hurwitz matrix

Linear complementarity problem

M-matrix