For details refer to http://en.wikipedia.org/wiki/Singular_value_decomposition. Here I want to highlight something.

Let $H$ be a square matrix, maybe not symmetric. Then

1. In general, its singular values have no relationship with its eigenvalues!!

2. It’s singular values are related with the eigenvalues of $HH^T or H^TH$

The non-zero singular values of $H$ are the square roots of the non-zero eigenvalues of $H^TH or HH^T$. Note $H^TH$ and \$latex  HH^T\$ are transpose to each other, they have the same eigenvalues!

3. For a symmetric matrix, its eigenvalues are the same as its singular values? Partly wrong!

Note if the eigenvalues can be negtive, while the singular values are positive. So for a symmetric matrix, its eigenvalues and singular values have the same absolute values. The positive eigenvalues are the same as singular values.