P-matrix
In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of [math]\displaystyle{ P_0 }[/math]-matrices, which are the closure of the class of P-matrices, with every principal minor [math]\displaystyle{ \geq }[/math] 0.
Spectra of P-matrices
By a theorem of Kellogg,[1][2] the eigenvalues of P- and [math]\displaystyle{ P_0 }[/math]- matrices are bounded away from a wedge about the negative real axis as follows:
- If [math]\displaystyle{ \{u_1,...,u_n\} }[/math] are the eigenvalues of an n-dimensional P-matrix, where [math]\displaystyle{ n\gt 1 }[/math], then
- [math]\displaystyle{ |\arg(u_i)| \lt \pi - \frac{\pi}{n},\ i = 1,...,n }[/math]
- If [math]\displaystyle{ \{u_1,...,u_n\} }[/math], [math]\displaystyle{ u_i \neq 0 }[/math], [math]\displaystyle{ i = 1,...,n }[/math] are the eigenvalues of an n-dimensional [math]\displaystyle{ P_0 }[/math]-matrix, then
- [math]\displaystyle{ |\arg(u_i)| \leq \pi - \frac{\pi}{n},\ i = 1,...,n }[/math]
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 [math]\displaystyle{ \mathrm{LCP}(M,q) }[/math] 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 [math]\displaystyle{ \mathbb{R}^n }[/math].[5]
A related class of interest, particularly with reference to stability, is that of [math]\displaystyle{ P^{(-)} }[/math]-matrices, sometimes also referred to as [math]\displaystyle{ N-P }[/math]-matrices. A matrix A is a [math]\displaystyle{ P^{(-)} }[/math]-matrix if and only if [math]\displaystyle{ (-A) }[/math] is a P-matrix (similarly for [math]\displaystyle{ P_0 }[/math]-matrices). Since [math]\displaystyle{ \sigma(A) = -\sigma(-A) }[/math], 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. 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". Linear Algebra and its Applications 5 (1): 65–108. doi:10.1016/0024-3795(72)90019-5. 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.
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. http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf.
- 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 [math]\displaystyle{ P_0 }[/math]-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)
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