P-matrix

In mathematics, a P-matrix is a complex square matrix with every principal minor > 0. A closely related class is that of P_0-matrices, which are the closure of the class of P-matrices, with every principal minor \geq 0.

Spectra of P-matrices

By a theorem of Kellogg,[1][2] the eigenvalues of P- and P_0- matrices are bounded away from a wedge about the negative real axis as follows:

If \{u_1,...,u_n\} are the eigenvalues of an n-dimensional P-matrix, then
|arg(u_i)| < \pi - \frac{\pi}{n}, i = 1,...,n
If \{u_1,...,u_n\}, u_i \neq 0, i = 1,...,n are the eigenvalues of an n-dimensional P_0-matrix, then
|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 LCP(M,q) 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 \mathbb{R}^n.[5]

A related class of interest, particularly with reference to stability, is that of P^{(-)}-matrices, sometimes also referred to as N-P-matrices. A matrix A is a P^{(-)}-matrix if and only if (-A) is a P-matrix (similarly for P_0-matrices). Since \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". Linear Algebra and its Applications 5 (1): 65–108. doi:10.1016/0024-3795(72)90019-5.
  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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