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Singular Value and Eigenvalue

04/04/2011

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.

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