Characterizing and Testing Principal Minor Equivalence of Matrices

TL;DR

This paper characterizes principal minor equivalence of matrices and provides a polynomial-time algorithm to determine if two matrices are equivalent, with applications to determinantal point processes and polynomial identity testing.

Contribution

It introduces a complete characterization of principal minor equivalence and develops a deterministic polynomial-time algorithm for testing equivalence.

Findings

Polynomial-time algorithm for principal minor equivalence.

Application to checking equality of determinantal point processes.

Application to polynomial identity testing with determinant representations.

Abstract

Two matrices are said to be principal minor equivalent if they have equal corresponding principal minors of all orders. We give a characterization of principal minor equivalence and a deterministic polynomial time algorithm to check if two given matrices are principal minor equivalent. Earlier such results were known for certain special cases like symmetric matrices, skew-symmetric matrices with {0, 1, -1}-entries, and matrices with no cuts (i.e., for any non-trivial partition of the indices, the top right block or the bottom left block must have rank more than 1). As an immediate application, we get an algorithm to check if the determinantal point processes corresponding to two given kernel matrices (not necessarily symmetric) are the same. As another application, we give a deterministic polynomial-time test to check equality of two multivariate polynomials, each computed by a symbolic determinant with a rank 1 constraint on coefficient matrices.