P-matrix

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.

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 $\{u_1,...,u_n\}$ are the eigenvalues of an $n$-dimensional $P$-matrix, 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

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

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 $LCP(M,q)$ has a unique solution for every vector $q$ if and only if $M$ is a $P$-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.

See also

• Q-matrix
• Z-matrix (mathematics)
• M-matrix
• Perron–Frobenius theorem
• Hurwitz matrix
• Linear complementarity problem

References

• David Gale and Hukukane Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann. 159:81-93 (1965) doi:10.1007/BF01360282
• 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)