P-matrix

In mathematics, a -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 -matrices, with every principal minor $\geq$ 0.

Spectra of -matrices

By a theorem of Kellogg, the eigenvalues of - and $P_0$- matrices are bounded away from a wedge about the negative real axis as follows:

:If $\$ are the eigenvalues of an -dimensional -matrix, where $n>1$, then
::$, \arg(u_i), < \pi - \frac,\ i = 1,...,n$
:If $\$, $u_i \neq 0$, $i = 1,...,n$ are the eigenvalues of an -dimensional $P_0$-matrix, then
::$, \arg(u_i), \leq \pi - \frac,\ i = 1,...,n$

Remarks

The class of nonsingular ''M''-matrices is a subset of the class of -matrices. More precisely, all matrices that are both -matrices and ''Z''-matrices are nonsingular -matrices. The class of sufficient matrices is another generalization of -matrices.

The linear complementarity problem $\mathrm(M,q)$ has a unique solution for every vector if and only if is a -matrix. This implies that if is a -matrix, then is a -matrix.

If the Jacobian of a function is a -matrix, then the function is injective on any rectangular region of $\mathbb^n$.

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 is a $P^$-matrix if and only if $(-A)$ is a -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

Notes

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 - and $P_0$-Matrices, ''Linear Algebra and its Applications'' 119:1-25 (1989)
* R. B. Kellogg, On complex eigenvalues of and {{mvar, P matrices, ''Numer. Math.'' 19:170-175 (1972)

Matrix theory
Matrices (mathematics)