A stability result using the matrix norm to bound the permanent

Abstract

We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose A is an n × n matrix over C (resp. R), and let P denote the set of n × n matrices over C (resp. R) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of A satisfies |per(A)| ≤ ||A||ⁿ₂ with equality iff A/||A||₂ ∈ P (where ||A||₂ is the operator 2-norm of A). We show a stability version of this result asserting that unless A is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of n) than ||A||ⁿ₂. In particular, for any fixed α, β > 0, we show that |per(A)| is exponentially smaller than ||A||ⁿ₂ unless all but at most αn rows contain entries of modulus at least ||A||₂(1−β).