Reducing Complex Matrices to Condensed Forms by Unitary Congruence Transformations

We show that any $n\times n$ conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding $1,2,3,\ldots$, respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix.