Linear Time Interactive Certificates for the Minimal Polynomial and the Determinant of a Sparse Matrix

TL;DR

This paper introduces efficient algorithms with certificates for verifying the minimal polynomial, determinant, and characteristic polynomial of sparse matrices, using linear time verification and minimal randomness.

Contribution

It presents a novel linear-time certificate algorithm for the minimal polynomial and determinant of sparse matrices, along with a new preconditioner ensuring irreducibility of the characteristic polynomial.

Findings

Verification complexity requires only one matrix-vector multiplication

Preconditioner uses only two random entries and is applied in linear time

Certificates enable efficient probabilistic verification of matrix invariants

Abstract

Computational problem certificates are additional data structures for each output, which can be used by a-possibly randomized-verification algorithm that proves the correctness of each output. In this paper, we give an algorithm that computes a certificate for the minimal polynomial of sparse or structured nxn matrices over an abstract field, of sufficiently large cardinality, whose Monte Carlo verification complexity requires a single matrix-vector multiplication and a linear number of extra field operations. We also propose a novel preconditioner that ensures irreducibility of the characteristic polynomial of the generically preconditioned matrix. This preconditioner takes linear time to be applied and uses only two random entries. We then combine these two techniques to give algorithms that compute certificates for the determinant, and thus for the characteristic polynomial, whose Monte Carlo verification complexity is therefore also linear.