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Heavytortoise
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For a square, complex-symmetric matrix ##A##, the columns of the right and left matrices ##U## and ##V## of the singular value decomposition should be complex conjugates, since for [tex]A=A^T, A\in{\mathbb C}^{N\times N}[/tex],
[tex]
A = U\Sigma V^H, A^T=(U\Sigma V^H)^T
[/tex]
so that
[tex]
U\Sigma V^H=(V^H)^T\Sigma U^T.
[/tex]
Then we have [tex]U=(V^H)^T[/tex], right? So why isn't this the case when I run a few experiments with Matlab? The magnitudes of the elements of ##U## and ##V## are equal, but they aren't conjugates. The expected relationship holds for real ##A##, where ##U## and ##V## are real-valued, but not for complex symmetric matrices. Who's screwed up here, me or Matlab?
[tex]
A = U\Sigma V^H, A^T=(U\Sigma V^H)^T
[/tex]
so that
[tex]
U\Sigma V^H=(V^H)^T\Sigma U^T.
[/tex]
Then we have [tex]U=(V^H)^T[/tex], right? So why isn't this the case when I run a few experiments with Matlab? The magnitudes of the elements of ##U## and ##V## are equal, but they aren't conjugates. The expected relationship holds for real ##A##, where ##U## and ##V## are real-valued, but not for complex symmetric matrices. Who's screwed up here, me or Matlab?
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