Geodesic
A Unitary Relation Between a Matrix and its Transpose
Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 279-280

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It is well known that if A is an n × n complex matrix and A T is its transpose, then there is an invertible n x n complex matrix S such that A T = S -1 AS. In this note we wish to point out another simple relation between A and A T .If A is an n × n complex matrix and A T is its transpose then there are unitary n × n complex matrices U and V such that A T = UAV. 
Gordon, William R. A Unitary Relation Between a Matrix and its Transpose. Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 279-280. doi: 10.4153/CMB-1970-056-7
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[1] 1. Eckart, C. and Young, G., A principal axis transformation for non-hermitian matrices. Bull. Amer. Math. Soc. 45 (1939), 118-121. Google Scholar

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