P-matrix

By a theorem of Kellogg,{{cite journal|last1=Kellogg|first1=R. B.|title=On complex eigenvalues ofM andP matrices|journal=Numerische Mathematik|date=April 1972|volume=19|issue=2|pages=170–175|doi=10.1007/BF01402527}}{{cite journal|last1=Fang|first1=Li|title=On the spectra of P- and P0-matrices|journal=Linear Algebra and Its Applications|date=July 1989|volume=119|pages=1–25|doi=10.1016/0024-3795(89)90065-7|doi-access=free}} the eigenvalues of {{mvar|P}}- and $P\_0$- matrices are bounded away from a wedge about the negative real axis as follows:

:If $\backslash \{u\_1,...,u\_n\backslash \}$ are the eigenvalues of an {{mvar|n}}-dimensional {{mvar|P}}-matrix, where $n>1$, then

::

:If $\backslash \{u\_1,...,u\_n\backslash \}$, $u\_i\; \backslash neq\; 0$, $i\; =\; 1,...,n$ are the eigenvalues of an {{mvar|n}}-dimensional $P\_0$-matrix, then

::$|\backslash arg(u\_i)|\; \backslash leq\; \backslash pi\; -\; \backslash frac\{\backslash pi\}\{n\},\backslash \; i\; =\; 1,...,n$

The class of nonsingular M-matrices is a subset of the class of {{mvar|P}}-matrices. More precisely, all matrices that are both {{mvar|P}}-matrices and Z-matrices are nonsingular {{mvar|M}}-matrices. The class of sufficient matrices is another generalization of {{mvar|P}}-matrices.{{cite journal|first1=Zsolt|last1=Csizmadia|first2=Tibor|last2=Illés|title=New criss-cross type algorithms for linear complementarity problems with sufficient matrices|journal=Optimization Methods and Software|volume=21|year=2006|number=2|pages=247–266|doi=10.1080/10556780500095009|

url=http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf|mr=2195759}}

The linear complementarity problem $\backslash mathrm\{LCP\}(M,q)$ has a unique solution for every vector {{mvar|q}} if and only if {{mvar|M}} is a {{mvar|P}}-matrix.{{cite journal|last1=Murty|first1=Katta G.|title=On the number of solutions to the complementarity problem and spanning properties of complementary cones|journal=Linear Algebra and Its Applications|date=January 1972|volume=5|issue=1|pages=65–108|doi=10.1016/0024-3795(72)90019-5|url=https://deepblue.lib.umich.edu/bitstream/2027.42/34188/1/0000477.pdf|hdl=2027.42/34188|hdl-access=free}} This implies that if {{mvar|M}} is a {{mvar|P}}-matrix, then {{mvar|M}} is a Q-matrix.

If the Jacobian of a function is a {{mvar|P}}-matrix, then the function is injective on any rectangular region of $\backslash mathbb\{R\}^n$.{{cite journal|last1=Gale|first1=David|last2=Nikaido|first2=Hukukane|title=The Jacobian matrix and global univalence of mappings|journal=Mathematische Annalen|date=10 December 2013|volume=159|issue=2|pages=81–93|doi=10.1007/BF01360282}}

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 {{mvar|A}} is a $P^\{(-)\}$-matrix if and only if $(-A)$ is a {{mvar|P}}-matrix (similarly for $P\_0$-matrices). Since $\backslash sigma(A)\; =\; -\backslash sigma(-A)$, the eigenvalues of these matrices are bounded away from the positive real axis.

== See also ==

• Routh–Hurwitz matrix
• Linear complementarity problem
• M-matrix
• Q-matrix
• Z-matrix
• Perron–Frobenius theorem

• {{cite journal|first1=Zsolt|last1=Csizmadia|first2=Tibor|last2=Illés|title=New criss-cross type algorithms for linear complementarity problems with sufficient matrices|journal=Optimization Methods and Software|volume=21|year=2006|number=2|pages=247–266|doi=10.1080/10556780500095009|

url=http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf|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 {{mvar|P}}- and $P\_0$-Matrices, Linear Algebra and its Applications 119:1-25 (1989)
• R. B. Kellogg, On complex eigenvalues of {{mvar|M}} and {{mvar|P}} matrices, Numer. Math. 19:170-175 (1972)

Category:Matrix theory

Category:Matrices (mathematics)