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P-matrix

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Complex square matrix for which every principal minor is positive

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

Spectra of P-matrices

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

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

The linear complementarity problem L C P ( M , q ) {\displaystyle \mathrm {LCP} (M,q)} has a unique solution for every vector q if and only if M is a P-matrix. This implies that if M is a P-matrix, then M is a Q-matrix.

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

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

See also

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" (PDF). Linear Algebra and Its Applications. 5 (1): 65–108. doi:10.1016/0024-3795(72)90019-5. hdl:2027.42/34188.
  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

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