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Characterizing and Testing Principal Minor Equivalence of Matrices

Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, Roshan Raj

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.

Characterizing and Testing Principal Minor Equivalence of Matrices

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.
Paper Structure (28 sections, 21 theorems, 123 equations, 2 figures, 2 algorithms)

This paper contains 28 sections, 21 theorems, 123 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The cut-transpose operation.
  3. Applications
  4. Polynomial Identity Testing.
  5. Determinantal Point Processes.
  6. Proof overview
  7. Reduction to the case of all nonzero entries.
  8. No common cuts.
  9. Base case: $4 \times 4$ matrices
  10. Getting a common cut.
  11. Decomposition into smaller matrices.
  12. An efficient algorithm.
  13. Applications to PIT.
  14. Notation and Preliminaries
  15. Principal minor equivalence
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $A$ and $B$ be two $n\times n$ irreducible matrices over any field. Then, $A$ and $B$ are principal minor equivalent if and only if there exists a sequence of $n\times n$ matrices $(A=A_0,A_1,\dots, A_k)$ with $k< 2n$ such that and $A_k$ is diagonally equivalent to $B$.

Figures (2)

  • Figure 1: Directed graphs associated with two matrices $A$ and $B$.
  • Figure 2: Applying cut-transpose on the directed graph associated with matrix $B$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Definition 2.2: Reducible and Irreducible matrix
  • Lemma 2.4
  • Definition 2.5: Cut of a matrix
  • Lemma 2.6
  • Claim 2.7
  • proof : Proof Sketch
  • ...and 39 more