P-matrix

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In mathematics, a $P$ -matrix is a complex square matrix with every principal minor > 0. 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.

[edit] 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:

{u 1,..., u n}

are the eigenvalues of an

n

-dimensional

P

-matrix, then

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

{u 1,..., u n}

, $u_i \neq 0$ ,

i = 1,..., n

are the eigenvalues of an

n

-dimensional

P 0

-matrix, then

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

[edit] Notes

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.

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 $σ(A) = - σ( - A)$ , the eigenvalues of these matrices are bounded away from the positive real axis.

[edit] References

R. B. Kellogg, On complex eigenvalues of $M$ and $P$ matrices, Numer. Math. 19:170-175 (1972)
Li Fang, On the Spectra of $P$ - and $P 0$ -Matrices, Linear Algebra and its Applications 119:1-25 (1989)
D. Gale and H. Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann. 159:81-93 (1965)