# A stability result using the matrix norm to bound the permanent

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## 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*||^{ n }_{2} 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*||^{ n }_{2}. In particular, for any fixed α, β > 0, we show that |per(*A*)| is exponentially smaller than ||*A*||^{ n }_{2} unless all but at most *αn* rows contain entries of modulus at least ||*A*||_{2}(1−*β*).

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