The fibre of P-matrices: the recursive construction of all matrices with positive principal minors

Abstract

P-matrices have positive principal minors and include many well-known matrix classes (positive definite, totally positive, M-matrices, etc.) How does one construct a generic P-matrix? Specifically, is there a characterization of P-matrices that lends itself to the tractable construction of every P-matrix? To answer these questions positively, a recursive method is employed that is based on a characterization of rank-one perturbations that preserve the class of P-matrices.

COMMUNICATED BY:

• R. Loewy

Keywords:

• P-matrix
• principal minors
• rank-one perturbation

AMS Subject Classifications:

• 15A48
• 15A15
• 15A23
• 68Q15

Acknowledgments

The authors acknowledge with thanks the comments provided by the Handling Editor and an anonymous referee, which helped improve our results and clarify our efforts.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Recall that the LCP defined by $A\in M_n\left(\mathbb{R}\right)$ and a vector $q\in {\mathbb{R}}^n$, is the problem of finding (entrywise) nonnegative vectors x and y such that y=Ax+q and $x^{T}y=0$.

2 $B=\left[b_{ij}\right]\in M_n\left(\mathbb{C}\right)$ is strictly row diagonally dominant if $|b_{ii}|>\sum _{j\ne i}|b_{ij}|$ ($i=1,2,\dots ,n$).