# P-matrix

In mathematics, a **P-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 P-matrices, with every principal minor 0.

## Spectra of P-matrices

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

- If are the eigenvalues of an n-dimensional P-matrix, where , then
- If , , are the eigenvalues of an n-dimensional -matrix, then

## 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 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 .[5]

A related class of interest, particularly with reference to stability, is that of -matrices, sometimes also referred to as -matrices. A matrix A is a -matrix if and only if is a P-matrix (similarly for -matrices). Since , the eigenvalues of these matrices are bounded away from the positive real axis.

## See also

## 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" (PDF).
*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 -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)