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 ${\displaystyle P_{0}}$-matrices, which are the closure of the class of P-matrices, with every principal minor ${\displaystyle \geq }$ 0.

Spectra of P-matrices

By a theorem of Kellogg,^[1][2] the eigenvalues of P- and ${\displaystyle P_{0}}$- matrices are bounded away from a wedge about the negative real axis as follows:

If ${\displaystyle \{u_{1},...,u_{n}\}}$ are the eigenvalues of an n-dimensional P-matrix, where ${\displaystyle n>1}$, then

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

If ${\displaystyle \{u_{1},...,u_{n}\}}$, ${\displaystyle u_{i}\neq 0}$, ${\displaystyle i=1,...,n}$ are the eigenvalues of an n-dimensional ${\displaystyle P_{0}}$-matrix, then

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

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.^[3]

The linear complementarity problem ${\displaystyle \mathrm {LCP} (M,q)}$ has a unique solution for every vector q if and only if M is a P-matrix.^[4] 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 ${\displaystyle \mathbb {R} ^{n}}$.^[5]

A related class of interest, particularly with reference to stability, is that of ${\displaystyle P^{(-)}}$-matrices, sometimes also referred to as ${\displaystyle N-P}$-matrices. A matrix A is a ${\displaystyle P^{(-)}}$-matrix if and only if ${\displaystyle (-A)}$ is a P-matrix (similarly for ${\displaystyle P_{0}}$-matrices). Since ${\displaystyle \sigma (A)=-\sigma (-A)}$, the eigenvalues of these matrices are bounded away from the positive real axis.

See also

• Routh–Hurwitz matrix
• Linear complementarity problem
• M-matrix
• Q-matrix
• Z-matrix
• Perron–Frobenius theorem