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

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

If $\{u_1,\mathrm{...},u_n\}$, $u_i\ne 0$, $i=1,\mathrm{...},n$ are the eigenvalues of an n-dimensional $P_0$-matrix, then

$|\arg (u_i)|\le \pi -\frac{\pi }{n},i=1,\mathrm{...},n$

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 $\mathrm{L}\mathrm{C}\mathrm{P}(M,q)$ 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 ${ℝ}^n$.

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.

See also

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

References

• David Gale and Hukukane Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann. 159:81-93 (1965)
• Li Fang, On the Spectra of P- and $P_0$-Matrices, Linear Algebra and its Applications 119:1-25 (1989)
• R. B. Kellogg, On complex eigenvalues of M and P matrices, Numer. Math. 19:170-175 (1972)