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,^[1]^[2] 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

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.^[3]

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

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

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.

Notes

↑ Kellogg, R. B. (April 1972). "On complex eigenvalues ofM andP matrices". Numerische Mathematik. 19 (2): 170–175. doi:10.1007/BF01402527.
↑ 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.
↑ 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.
↑ 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.
↑ 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.

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.
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)

This article is issued from Wikipedia - version of the 6/30/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

P-matrix

Spectra of -matrices

Remarks

See also

Notes

References

Spectra of $P$ -matrices