Bounds on minors of binary matrices

TL;DR

This paper establishes a simplified proof for an upper bound on the sums of squares of minors in {+1, -1} matrices, with implications for Hadamard matrices and determinants of random matrices.

Contribution

It provides a simpler proof of an existing bound on minors of {+1, -1} matrices and extends results related to Hadamard matrices and random matrix determinants.

Findings

Bound is sharp for Hadamard matrices

Simplified proof of the sum of squares minors bound

Generalization of Turan's determinant result

Abstract

We prove an upper bound on sums of squares of minors of {+1, -1} matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin (2009), but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices, and generalise a result of Turan on determinants of random {+1,-1} matrices.