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

Abstract

Let F n 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 = ( c 1 …, c n ) t ∈ F n , define the c-norm of x = ( x 1,…, x n ) t ∈ F n by | x | c = max ⁡ { | c t P x | : P ∈ GP ( n ) } , and the induced c-norm of A ∈ F n × n by | | A | | c = max ⁡ { | A y | c : | y | c ⩽ 1 } . We characterize, the isometries for induced c-norms. To achieve our goal, we study the geometric properties of F n × n, and obtain some inequalities related to the induced c-norm as by-products. These results are of independent interest.