[Math] Inequality for singular value for differences of matrices (upper bound)

Question

Does anybody know the inequality of singular value for differences of matrices, i.e.

$\sigma_{max}\left(\begin{array}{c}
A-B\end{array}\right)\leq??$

in term of $\sigma_{max}\left(\begin{array}{c}
A\end{array}\right)$, $\sigma_{min}\left(\begin{array}{c}
A\end{array}\right)$, $\sigma_{max}\left(\begin{array}{c}
B\end{array}\right)$, or $\sigma_{min}\left(\begin{array}{c}
B\end{array}\right)$

A and B are not Hermitian though.

Answer 1

Recall that $\sigma_\max(M)=\max_{\|x\|=1}\|Mx\|$. By triangular inequality, clearly we have $\sigma_\max(A-B)\le\sigma_\max(A)+\sigma_\max(B)$.