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Journal of Operator Theory

Volume 68, Issue 1, Summer 2012  pp. 179-203.

Unitary equivalence of a matrix to its transpose

Authors:  Stephan Ramon Garcia (1) and James E. Tener (2)
Author institution: (1) Department of Mathematics, Pomona College, Claremont, California, 91711, U.S.A.
(2) Department of Mathematics, University of California, Berkeley, California, 94721, U.S.A.

Summary:  Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the na\"ive assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix holds for matrices $7 \times 7$ and smaller, but fails for matrices $8 \times 8$ and larger.

Keywords:  Complex symmetric matrix, complex symmetric operator, unitary orbit, unitary equivalence, linear preserver, skew-Hamiltonian, numerical range, transpose, UECSM

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