Hi,
Given a complex unitary matrix $U$, can we find a real orthogonal matrix $K$ such that the product $KU$ is a complex symmetric matrix.
Thanks,
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2$\begingroup$ Have you tried just playing with the $2\times 2$ cases? $\endgroup$– Willie WongCommented Apr 11, 2012 at 15:40
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$\begingroup$ yes, but i could not see the results even in that case. $\endgroup$– zatilokumCommented Apr 12, 2012 at 14:42
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$\begingroup$ i need this result to see that if u is a complex unitary matrix, then $U=K.exp(iX)$ for $K$ is a real orthogonal matrix and $X$ is a real symmetric matrix. This is called polar decomposition for unitary matrices. For n=2, ı know that every symmetric unitary matrix can be written of the form $exp(iX)$ where $X$ is real symmetric. $\endgroup$– zatilokumCommented Apr 13, 2012 at 12:41
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1 Answer
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$U^TU$ is symmetric and unitary, so there exists a symmetric unitary $V$ such that $V^2=U^TU$. Set $K=VU^{-1}$, then $K$ is unitary and $K^TK=I$. Therefore, $K$ is real orthogonal as desired.
