Permutation invariant norms Original Research Article

A norm or seminorm on short parallel · short parallel on imagen is permutation invariant if short parallelPxshort parallel = short parallelxshort parallel for all n × n permutation matrices P and for all x set membership, variant Rn. We present a systematic study of permutation invariant seminorms. Their relations with other types of norms on matrices are discussed. In addition, we consider a special class of permutation invariant seminorms, the c-radii, defined and denoted by Rc(x) = max{ctPx : P a permutation} for any given c set membership, variantRn. It is shown that Rc are the building blocks of all permutation invariant seminorms. If the entries of c are not all equal and their sum is not 0, then Rc is a norm on imagen. For such Rc, we study their norm properties and characterize their isometry groups.