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, where $n>1$, 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

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

References

• Csizmadia, Zsolt; Illés, Tibor (2006). "New criss-cross type algorithms for linear complementarity problems with sufficient matrices" (pdf). Optimization Methods and Software. 21 (2): 247–266. doi:10.1080/10556780500095009. MR 2195759. {{cite journal}}: Invalid |ref=harv (help)
• 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)