Linear transformations on that preserve the Ky Fank-norm and a remarkable special case when (nk) = (4, 2)

Abstract

Given an n × n real matrix A. the Ky Fan k-norm of it is defined as the sum of its k largest singular values. We determine the structures of the linear transformations T on that preserve the Ky Fan k-norm. We show that except for (n,k) = (4,2), there exist orthogonal matrices U and V such that where A+ either denotes A or denotes At . For the particular case (n,k) = (4, 2)T is either of the usual form mentioned above or equivalent to the composition of the usual form with a particular operator that has many remarkable properties.