Let $A \in \mathbb{R}^{n\times m}$. Define
$$\|A\| = \sum_{i=1}^{\min(n,m)} \sigma_i$$
where $\sigma_i$ are the singular values of $A$. How to show that $\|A\|$ is norm?
We want to verify the definition of norm.
Homogeneity $\|\alpha A\| = |\alpha| \times \|A\|$ for $\alpha \in \mathbb{R}$
Triangular Inequality: $\|A+B\| \leq \|A\| + \|B\|$
Positive definiteness: if $\|A\|=0$, then $A=0$