Thanks – I have fixed the typo.

]]>The trace must be missing the superscript $trace(V^*A^*AV)$ after the 3rd equal sign below “For the Frobenius norm, using” ]]>

Dear professor Nick, thank you for your reply. I have been trying to understand the fractional powers of unitaries. I know you had cited a few references in the post. It seems that these references mostly talk about these matrices in a more general sense. It would be nice to see if any other papers/articles that deal with unitary cases rigorously.

]]>A^\alpha as defined here is unique even for unitary matrices: the (principal) log in the definition takes care of phase freedom. So unitary matrices are not special as regards fractional powers.

]]>Ah, perfect! Thank you for the fast reply.

]]>This case is easier as there is no constraint on the diagonal elements.

The problem is now to find the nearest positive semidefinite matrix.

See

Diagonally Perturbing a Symmetric Matrix to Make It Positive Definite

]]>

Thanks you for the great blog posts.

I am currently facing a nearest *covariance* problem, rather than correlation. I have searched through many of your papers / Nag documentation, but couldn’t see any advice on this.

Is there a simple way of adapting the solutions from the correlation setting?

Thank you.

]]>I see. Thanks for your time!

]]>The reference for Theorem 3 is Stanley C. Eisenstat and Ilse C. F. Ipsen,Relative Perturbation Techniques for Singular Value Problems,https://epubs.siam.org/doi/10.1137/0732088. If you want to cite this blog post you can use this BibTeX entry:

@misc{high21s,

author = “Nicholas J. Higham”,

title = “Singular Value Inequalities”,

year = 2021,

month = may,

howpublished = “\url{https://nhigham.com/2021/05/04/singular-value-inequalities/}”

}