Zero patterns and unitary similarity

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 LI(n). A pattern I is universal if every matrix in L(n) is unitarily similar to some matrix in LI(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.