P-matrix

Short description: Complex square matrix for which every principal minor is positive

In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of [math]\displaystyle{ P_0 }[/math]-matrices, which are the closure of the class of P-matrices, with every principal minor [math]\displaystyle{ \geq }[/math] 0.

Spectra of P-matrices

By a theorem of Kellogg, the eigenvalues of P- and [math]\displaystyle{ P_0 }[/math]- matrices are bounded away from a wedge about the negative real axis as follows:

If [math]\displaystyle{ \{u_1,...,u_n\} }[/math] are the eigenvalues of an n-dimensional P-matrix, where [math]\displaystyle{ n\gt 1 }[/math], then

[math]\displaystyle{ |\arg(u_i)| \lt \pi - \frac{\pi}{n},\ i = 1,...,n }[/math]

If [math]\displaystyle{ \{u_1,...,u_n\} }[/math], [math]\displaystyle{ u_i \neq 0 }[/math], [math]\displaystyle{ i = 1,...,n }[/math] are the eigenvalues of an n-dimensional [math]\displaystyle{ P_0 }[/math]-matrix, then

[math]\displaystyle{ |\arg(u_i)| \leq \pi - \frac{\pi}{n},\ i = 1,...,n }[/math]

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.

The linear complementarity problem [math]\displaystyle{ \mathrm{LCP}(M,q) }[/math] has a unique solution for every vector q if and only if M is a P-matrix. 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 [math]\displaystyle{ \mathbb{R}^n }[/math].

A related class of interest, particularly with reference to stability, is that of [math]\displaystyle{ P^{(-)} }[/math]-matrices, sometimes also referred to as [math]\displaystyle{ N-P }[/math]-matrices. A matrix A is a [math]\displaystyle{ P^{(-)} }[/math]-matrix if and only if [math]\displaystyle{ (-A) }[/math] is a P-matrix (similarly for [math]\displaystyle{ P_0 }[/math]-matrices). Since [math]\displaystyle{ \sigma(A) = -\sigma(-A) }[/math], 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

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. http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf. 
• 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 [math]\displaystyle{ P_0 }[/math]-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)