For a matrix $A \in \mathbb{C}^{n \times p}$, let $\sigma_1(A) \geq \sigma_{2}(A), \cdots, \sigma_{p}(A) \geq 0$ denote the singular values of $A$.
Since the operator norm of is just the largest singular value (i.e. $\sigma_1(A) = \| A \|_2$), we have that for matrices $A,B$, $\sigma_1(A+B) \leq \sigma_1(A) + \sigma_1(B)$.
Is this triangle inequality true for any other singular values as well? I think I found a counter example for the second largest.
Is there something interesting to say about it besides just "no"?
Best Answer
In general, we always have $\sigma_k(A+B)\leq \sigma_k(A) + \sigma_1(B)$ for any $k=1,\dots, p$. In fact taking $k=1$ recovers the inequality that you have.