Moving zeros among matrices

We investigate the zero-patterns that can be created by unitary similarity in a given matrix, and the zero-patterns that can be created by simultaneous unitary similarity in a given sequence of matrices. The latter framework allows a “simultaneous Hessenberg” formulation of Pati’s tridiagonal result for 4 × 4 matrices. This formulation appears to be a strengthening of Pati’s theorem. Our work depends at several points on the simplified proof of Pati’s result by Davidson and Djoković. The Hessenberg approach allows us to work with ordinary similarity and suggests an extension from the complex to arbitrary algebraically closed fields. This extension is achieved and related results for 5 × 5 and larger matrices are formulated and proved.