On tridiagonalization of matrices

AbstractWe consider the question: Is every n×n complex matrix unitarily similar to a tridiagonal one? It is shown that the answer is negative if n⩾6, and is affirmative if n=3. Additionally, some positive partial answers and related results are given. For example, (1) every pair of (Hermitian) projections is simultaneously unitarily similar to a pair of tridiagonal matrices; (2) if A–A∗ has a rank one, then A is unitarily similar to a tridiagonal matrix