Given an N-by-N asymmetric matrix $M$
Is there any theory about approximating $M$'s Ky Fan k-norm $|| M ||_k$ using $\frac {(M+M^T)}{2}$'s Ky Fan k-norm $|| \frac {(M+M^T)}{2} ||_k$?
UPDATE: Consider another symmetric matrix $S$
if $M=S+E$ can one approximate $||M||_k$ by $||S||_k$?
I doubt if there is any such approximation. Consider $M=\begin{pmatrix}0&\lambda\\-\lambda&0\end{pmatrix}\oplus0_{n-2}$ for instance. Its singular values are $1, 1, 0, \ldots, 0$ and hence its Ky-Fan $k$-norm is $\lambda$ when $k=1$ and $2\lambda$ when $k\ge2$. However, the singular values and and Ky-Fan $k$-norms of $(M+M^T)/2$ are constantly zero. You cannot generate any meaningful approximation of the Ky-Fan $k$-norms of $M$ from these zeroes.