On linear preservers of permanental rank

Let Mat(n)(F) denote the set of square n x n matrices over a field F of characteristic different from two. The permanental rank prk (A) of a matrix A is an element of Mat(n)(F) is the size of the maximal square submatrix in A with nonzero permanent. By Lambda(k) and A <= k we denote the subsets of matrices A is an element of Mat(n)(F) with prk (A) = k and prk (A) < k, respectively. In this paper for each 1 <= k <= n -1 we obtain a complete characterization of linear maps T : Mat(n)(F)-+ Mat(n)(F) satisfying T(Lambda (<= k)) = Lambda (<= k )or bijective linear maps satisfying T(Lambda (<= k)) C Lambda (<= k). Moreover, we show that if F is an infinite field, then Lambda(k) is Zariski dense in Lambda (<= k) and apply this to describe such bijective linear maps satisfying T(Lambda(k)) subset of Lambda(k).(c) 2023 Elsevier Inc. All rights reserved.