P-matrix

In mathematics, a P-matrix is a complex square matrix with every principal minor > 0. A closely related class is that of P₀-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, the eigenvalues of P- and P₀- matrices are bounded away from a wedge about the negative real axis as follows:

If {u₁,...,u_n} are the eigenvalues of an n-dimensional P-matrix, then

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

If {u₁,...,u_n}, $u_i \neq 0$, i = 1,...,n are the eigenvalues of an n-dimensional P₀-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.^[1]

If the Jacobian of a function is a P-matrix, then the function is injective on any rectangular region of $\mathbb{R}^n$.

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 P₀-matrices). Since σ(A) = − σ( − A), the eigenvalues of these matrices are bounded away from the positive real axis.

Notes

1. ^ 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. MR2195759. http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf. 

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. MR2195759. http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf. 
• David Gale and Hirafumi Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann. 159:81-93 (1965)
• Li Fang, On the Spectra of P- and P₀-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)