Relation between eigvals and singular vals.

[Pacopag]

Homework Statement

Is there a way to get the eigenvalues of a matrix from its singular values?

Homework Equations

Eigenvalues $$\lambda$$ satisfy $$Ax=\lambda x$$ where x are the eigenvectors.
Singular values $$\sigma$$ satisfy $$\sqrt{A^H A} v = \sigma v$$,
i.e. singular values of A are the eigenvalues of the matrix $$\sqrt{A^H A}$$

The Attempt at a Solution

It seems that
$$\lambda = \pm \sqrt{\sigma}$$
but this only gives some of the eigenvalues. How do I get the others?

 

[mighty2000]

Thank you for your question. Yes, there is a way to obtain the eigenvalues of a matrix from its singular values. As you have correctly stated, the singular values of a matrix are the square root of the eigenvalues of its Hermitian conjugate multiplied by the matrix itself. This means that the eigenvalues of the original matrix can be obtained by taking the square root of the singular values and multiplying them by the matrix.

However, it is important to note that the eigenvectors corresponding to these eigenvalues may not be the same as the eigenvectors of the original matrix. This is because the singular values are not unique and can be reordered or even duplicated. Therefore, to obtain the complete set of eigenvalues and eigenvectors of the original matrix, you will need to use other methods such as diagonalization or Jordan decomposition.

I hope this helps. Let me know if you have any further questions.