17/04/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Ilya Spitkovsky, College of William and Mary, Williamsburg, VA, USA
Matrices with normal defect one
A \(n\times n\) matrix \(A\) has normal defect one if it is not
normal, however can be embedded as a north-western block into a
normal matrix of size \((n+1)\times (n+1)\). The latter is called a
minimal normal completion of \(A\). A construction of all matrices
with normal defect one is given. Also, a simple procedure is
presented which allows one to check whether a given matrix has
normal defect one, and if this is the case, to construct all its
minimal normal completions. A characterization of the generic case
for each n under the assumption that the rank of the
self-commutator of \(A\) equals \(2\) (which is necessary for \(A\)
to have normal defect one) is obtained. Both the complex and the
real cases are considered. It is pointed out how these results can
be used to solve the minimal commuting completion problem in the
classes of pairs of \(n \times n\) Hermitian (resp., symmetric, or
symmetric/antisymmetric) matrices when the completed matrices are
sought of size \((n+1)\times (n+1)\). An application to the
\(2\times n\) separability problem in quantum computing is
described. This is a joint work with Dmitry Kaliuzhnyi-Verbovetskyi
and Hugo Wourdeman.