Simultaneous Unitary Similarity and Congruence

Start: 10/23/2013 - 4:15pm
End  : 10/23/2013 - 5:15pm


Roger A. Horn, University of Utah


Square complex matrices $ A $ and $ B $ are unitarily similar if there is a
unitary matrix $ U $ such that $ A=UBU^{\ast} $ (conjugate transpose); they are
unitarily congruent if there is a unitary $ U $ such that $ A=UBU^{T} $.

It is not difficult to determine whether a given pair of matrices is unitarily
similar. In fact, one can do so with finitely many ordinary arithmetic

However, the situation seems to be quite different for unitary congruence. The
best known result says that two given matrices are unitarily congruent if and
only if three specific pairs of matrices are simultaneously unitarily similar
(that is, the same $ U $ works for all three pairs). can we determine whether finitely many pairs of matrices are
simultaneously unitarily similar? What about simultaneous unitary congruence
of finitely many pairs of matrices?

Davidson Lecture Hall, Claremont McKenna College

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