Unitary Equivalence to a Complex Symmetric Matrix: Low Dimensions

A matrix T∈Mn(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above

Unitary Equivalence of a Matrix to Its Transpose

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