“P-matrix”的意思、由来-开放百科全书

In mathematics, a -matrix is a complex square matrix with every principal minor > 0. A closely related class is that of

-matrices, which are the closure of the class of

-matrices, with every principal minor

Spectra of

-matrices

By a theorem of Kellogg,^[1]^[2] the eigenvalues of

- and

- matrices are bounded away from a wedge about the negative real axis as follows:

Remarks

The class of nonsingular M-matrices is a subset of the class of

-matrices. More precisely, all matrices that are both

-matrices and Z-matrices are nonsingular

-matrices. The class of sufficient matrices is another generalization of

-matrices.^[3]

The linear complementarity problem

has a unique solution for every vector

if and only if

is a

-matrix.^[4]

If the Jacobian of a function is a

-matrix, then the function is injective on any rectangular region of

.^[5]

A related class of interest, particularly with reference to stability, is that of

-matrices, sometimes also referred to as

-matrices. A matrix

is a

-matrix if and only if

is a

-matrix (similarly for

-matrices). Since

, the eigenvalues of these matrices are bounded away from the positive real axis.

See also

Notes

1. ^{{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}}
2. ^{{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}}
3. ^{{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|format=pdf |mr=2195759|}}
4. ^{{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}}
5. ^{{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}}

References

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