P-matrix

In mathematics, a P-matrix is a complex square matrix with every principal minor > 0. A closely related class is that of ${\displaystyle P_{0}}$-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,[1][2] the eigenvalues of P- and ${\displaystyle P_{0}}$- matrices are bounded away from a wedge about the negative real axis as follows:

If ${\displaystyle \{u_{1},...,u_{n}\}}$ are the eigenvalues of an n-dimensional P-matrix, where ${\displaystyle n>1}$, then

${\displaystyle |\arg(u_{i})|<\pi -{\frac {\pi }{n}},\ i=1,...,n}$

If ${\displaystyle \{u_{1},...,u_{n}\}}$, ${\displaystyle u_{i}\neq 0}$, ${\displaystyle i=1,...,n}$ are the eigenvalues of an n-dimensional ${\displaystyle P_{0}}$-matrix, then

${\displaystyle |\arg(u_{i})|\leq \pi -{\frac {\pi }{n}},\ i=1,...,n}$

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 ${\displaystyle \mathrm {LCP} (M,q)}$ 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 ${\displaystyle \mathbb {R} ^{n}}$.[5]

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

See also

• Hurwitz matrix
• Linear complementarity problem
• M-matrix
• Q-matrix
• Z-matrix
• Perron–Frobenius theorem

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.
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

• 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 ${\displaystyle P_{0}}$-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)