Reachability of condensed forms via unitary congruence transformations

M. Ghasemi Kamalvand ; Kh. D. Ikramov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1339-1343

Résumé

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 similarity transformation; if $n\ge5$, then there exist matrices that cannot be brought to tridiagonal form by a unitary similarity transformation; for any fixed set of positions (pattern) $S$ whose cardinality exceeds $n(n-1)/2$, there exists an $n$-by-$n$ matrix $A$ such that none of the matrices that are unitarily similar to $A$ can have zeros in all of the positions in $S$. It is shown that analogous facts are valid if unitary similarity transformations are replaced by unitary congruence ones.