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In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. 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[edit]

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


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] 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 .[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[edit]


  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.