This is a follow-up question to [this one](https://mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums) about eigenvalues of matrix sums. Suppose you have matrices $A$ and $B$, and know their singular values. What can you say about the singular values of $A+B$?


For Hermitian matrices and eigenvalues, this question was answered by a famous theorem of [Knutson and Tao](http://www.ams.org/notices/200102/fea-knutson.pdf), but I don't know of anything similar for the more general case of singular values. This result would have come in useful for an estimate that I needed. I was able to obtain the estimate in a differenr way, but now I'm curious about the question.



Singular values of matrix sums
The singular values of a $n \times m$ matrix A are more or less the eigenvalues of the $n+m \times n+m$ matrix $\begin{pmatrix} 0 & A \\\ A^\* & 0 \end{pmatrix}$. By "more or less", I mean that one also has to throw in the negation of the singular values, as well as some zeroes. Using this, one can deduce inequalities for the singular values from that of the Hermitian matrices problem. This may even be the complete list of inequalities, though I don't know if this has already been proven in the literature.


See also my blog post on this at


<http://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/>


