  A subspace of the space, L(n), of traceless complex $n\times n$ 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 $n\le3$, and
partial results obtained for $n=4$. Two infinite families of universal patterns
are constructed. They give two analogues of Schur's triangularization theorem.
