On the permanental Congruence numerical range

Abstract

For with n ≥ 2 and 1 ≤ r ≤ n, let . In this note, we prove that if and only if A = {0} when r is even and if and only if A is skew symmetric when r is odd. We also prove that if and only if A is skew symmetric for 1 ≤ r ≤ n. Moreover, we show that is a circular disk centered at the origin if (i) A is symmetric (n ≥ 2) or A is skew symmetric (n > 2), when r = n; (ii) 1≥ r > n but when n is even. Moreover, is a circular disk centered at the origin for any with n ≥ 2 and 1 ≤ r ≤ n. We also discuss some generalization.