Skip to main content

[Community Question] Linear-algebra: Singular value inequality for sum of 2 matrices

One of our user asked:

I found a theorem mentioned in a couple of places, but could not find a proof. The theorem states the following:

Let $A, B \in \mathbb{F^{m,n}}$, $p=min(m,n)$ with singular values $\sigma_1(A) \geqslant...\geqslant \sigma_p(A)$ and $\sigma_i(B) \geqslant...\geqslant \sigma_p(B)$ respectively, then $\sigma_{i+j-1}(A+B) \leqslant \sigma_i(A) + \sigma_j(B)$.

I am looking for a proof of the above. Thanks in advance.


Comments

Popular posts from this blog

[Community Question] Linear-algebra: Explicit example of an additive map which is not R-linear

One of our user asked: Is there an explicit example of an additive map $\mathbb{R}^n \rightarrow \mathbb{R}^m$ which is not linear? (I have mostly thought about the question when $m = n = 1$ , and I don't think the general case is any easier.) I know that something like $f: \mathbb{C} \rightarrow \mathbb{C}$ which sends $f: z \mapsto \text{Real}(z)$ would be additive but not $\mathbb{C}$ -linear. I also know that since $\mathbb{R}$ is a $\mathbb{Q}$ -vector space, I can find some example where $1 \mapsto 1$ and $\sqrt{2} \mapsto 0$ . Is there an explicit example? By explicit, I mean, given an element in the domain, there would be some procedure to decide where it maps. Thank you!