# SVD with Matlab

1. Oct 8, 2007

### Heavytortoise

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 $$A=A^T, A\in{\mathbb C}^{N\times N}$$,
$$A = U\Sigma V^H, A^T=(U\Sigma V^H)^T$$
so that
$$U\Sigma V^H=(V^H)^T\Sigma U^T.$$
Then we have $$U=(V^H)^T$$, 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?