Singular Value and Eigenvalue
For details refer to http://en.wikipedia.org/wiki/Singular_value_decomposition. Here I want to highlight something.
Let 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
The non-zero singular values of are the square roots of the non-zero eigenvalues of . Note 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.