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,12 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.3
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.4 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}} .5
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
- Routh–Hurwitz matrix
- Linear complementarity problem
- M-matrix
- Q-matrix
- Z-matrix
- Perron–Frobenius theorem
Notes
- 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 {\displaystyle 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)
References
Kellogg, R. B. (April 1972). "On complex eigenvalues ofM andP matrices". Numerische Mathematik. 19 (2): 170–175. doi:10.1007/BF01402527. /wiki/Doi_(identifier) ↩
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. https://doi.org/10.1016%2F0024-3795%2889%2990065-7 ↩
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. http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf ↩
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. https://deepblue.lib.umich.edu/bitstream/2027.42/34188/1/0000477.pdf ↩
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. /wiki/Doi_(identifier) ↩