TITLE: minimize the sum of absolute eigenvalues
QUESTION [6 upvotes]: Hi,
We have a real symmetric matrix M with diagonal elements 0's, the eigenvalues and eigenvectors of M are computed.
Now we wish to change its diagonal elements arbitrarily to minimize the sum of absolute eigenvalues. Does there exist a way to find such modifications?
If we add a constraint : keep Tr(M)=0, would that become easier?
Is there some topics about these?
Thank you for your help
Zhi Ming

REPLY [3 votes]: Interesting question. Nuclear norm minimization is getting much attention right now as it relates directly to compressed sensing.
Some software for minimization with this constraint that I've used:
http://perception.csl.illinois.edu/matrix-rank/sample_code.html
A fun related problem:
http://www-stat.stanford.edu/~candes/papers/MatrixCompletion.pdf