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 ${\displaystyle P_{0}$-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

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)