Zero patterns and unitary similarity (Q984992)

Let \(M(n)\) be the algebra of \(n\times n\) complex matrices and \(L(n)\) be the subspace consisting of matrices of trace \(0\). The authors say that a subspace \(V\) of \(L(n)\) is universal if each matrix in \(L(n)\) is unitarily similar to a matrix in \(V\). Following the paradigm of Schur's theorem (every matrix in \(M(n)\) is unitarily similar to an upper triangular matrix) the authors investigate ``patterns'' \(I\subseteq\left\{ 1,\dots,n\right\} ^{2}\) for which the subspace \(V=L(n)_{I}\) consisting of matrices with entries equal to \(0\) at each index \((i,j)\in I\) is universal. Let \(\mathcal{K(}n)\) be the ideal generated by the elementary symmetric functions in \(x_{1},\dots,x_{n}\) in the polynomial ring \(\mathbb{R[}x_{1},\dots,x_{n}].\) A pattern \(I\) is defined to be singular if \({\prod_{(i,j)\in I}} (x_{i}-x_{j})\in\mathcal{K}(n)\). The authors prove that every nonsingular pattern \(I\) is universal and construct some explicit families of nonsingular (and hence universal) patterns. They give a complete description of universal patterns \(I\) for \(n\leq3\) and a partial description for \(n=4\).