Zero patterns and unitary similarity

Abstract

A subspace of the space, L ( n ) , of traceless complex n × n matrices can be specified by requiring that the entries at some positions ( i , j ) be zero. The set, I , of these positions is a (zero) pattern and the corresponding subspace of L ( n ) is denoted by L I ( n ) . A pattern I is universal if every matrix in L ( n ) is unitarily similar to some matrix in L I ( n ) . The problem of describing the universal patterns is raised, solved in full for n ⩽ 3 , and partial results obtained for n = 4 . Two infinite families of universal patterns are constructed. They give two analogues of Schur's triangularization theorem.