Isometries for the induced c-norm on square matrices and some related results

Let Fn be the linear space of column vectors with n coordinates over F = R or C. Denote by GP(n) the group of n × n generalized permutation matrices, i.e., matrices with exactly one nonzero entry with magnitude 1 in each row and column. Given a nonzero vector c = (c1…,cn)t Fn, define the c-norm of x = (x1,…,xn)t Fn by xc=max⁡{ctPx:P GP(n)}, and the induced c-norm of A Fn×n by Ac=max⁡{Ayc:yc 1}. We characterize, the isometries for induced c-norms. To achieve our goal, we study the geometric properties of Fn × n, and obtain some inequalities related to the induced c-norm as by-products. These results are of independent interest.