Reachability of Condensed Formsvia Unitary Congruence Transformations

摘要

There are several well-known facts about unitary similarity transformations of complex n-by-n matrices: every matrix of order n = 3 can be brought to tridiagonal form by a unitary similaritytransformation; if n≥5, then there exist matrices that cannot be brought to tridiagonal form by a unitary sim-ilarity transformation; for any fixed set of positions (pattern) S whose cardinality exceeds n(n – 1)/2, thereexists an n-by-n matrix A such that none of the matrices that are unitarily similar to A can have zeros inall of the positions in S. It is shown that analogous facts are valid if unitary similarity transformationsare replaced by unitary congruence ones.