P-matrix

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In mathematics, a ${\displaystyle P}$-matrix is a complex square matrix with every principal minor > 0. A closely related class is that of ${\displaystyle P_{0}}$-matrices, which are the closure of the class of ${\displaystyle P}$-matrices, with every principal minor ${\displaystyle \geq }$ 0.

Spectra of ${\displaystyle P}$-matrices

By a theorem of Kellogg,[1][2] the eigenvalues of ${\displaystyle P}$- and ${\displaystyle P_{0}}$- matrices are bounded away from a wedge about the negative real axis as follows:

If ${\displaystyle \{u_{1},...,u_{n}\}}$ are the eigenvalues of an ${\displaystyle n}$-dimensional ${\displaystyle P}$-matrix, where ${\displaystyle n>1}$, then
${\displaystyle |arg(u_{i})|<\pi -{\frac {\pi }{n}},i=1,...,n}$
If ${\displaystyle \{u_{1},...,u_{n}\}}$, ${\displaystyle u_{i}\neq 0}$, ${\displaystyle i=1,...,n}$ are the eigenvalues of an ${\displaystyle n}$-dimensional ${\displaystyle P_{0}}$-matrix, then
${\displaystyle |arg(u_{i})|\leq \pi -{\frac {\pi }{n}},i=1,...,n}$

Remarks

The class of nonsingular M-matrices is a subset of the class of ${\displaystyle P}$-matrices. More precisely, all matrices that are both ${\displaystyle P}$-matrices and Z-matrices are nonsingular ${\displaystyle M}$-matrices. The class of sufficient matrices is another generalization of ${\displaystyle P}$-matrices.[3]

The linear complementarity problem ${\displaystyle LCP(M,q)}$ has a unique solution for every vector ${\displaystyle q}$ if and only if ${\displaystyle M}$ is a ${\displaystyle P}$-matrix.[4]

If the Jacobian of a function is a ${\displaystyle P}$-matrix, then the function is injective on any rectangular region of ${\displaystyle \mathbb {R} ^{n}}$.[5]

A related class of interest, particularly with reference to stability, is that of ${\displaystyle P^{(-)}}$-matrices, sometimes also referred to as ${\displaystyle N-P}$-matrices. A matrix ${\displaystyle A}$ is a ${\displaystyle P^{(-)}}$-matrix if and only if ${\displaystyle (-A)}$ is a ${\displaystyle P}$-matrix (similarly for ${\displaystyle P_{0}}$-matrices). Since ${\displaystyle \sigma (A)=-\sigma (-A)}$, the eigenvalues of these matrices are bounded away from the positive real axis.

Notes

1. ^ Kellogg, R. B. (April 1972). "On complex eigenvalues ofM andP matrices". Numerische Mathematik. 19 (2): 170–175. doi:10.1007/BF01402527.
2. ^ Fang, Li (July 1989). "On the spectra of P- and P0-matrices". Linear Algebra and its Applications. 119: 1–25. doi:10.1016/0024-3795(89)90065-7.
3. ^ 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.
4. ^ Murty, Katta G. (January 1972). "On the number of solutions to the complementarity problem and spanning properties of complementary cones". Linear Algebra and its Applications. 5 (1): 65–108. doi:10.1016/0024-3795(72)90019-5.
5. ^ Gale, David; Nikaido, Hukukane (10 December 2013). "The Jacobian matrix and global univalence of mappings". Mathematische Annalen. 159 (2): 81–93. doi:10.1007/BF01360282.