Why do we use the rank of a matrix to compute the Schatten norms?

Question

The Schatten norm is defined as

$$\|A\|_{S_P} = \left(\sum_{i=1}^{r(A)}\sigma_i^P(A)\right)^{\frac{1}{P}}$$

where $r(A)$ represents the rank of the matrix $A$. Why do we use the rank of the matrix to compute the Schatten $p$-norm?

Answer 1

If you assume (as you do, although you don't say) that the singular values are ordered from bigger to smallest, you have $$ \sum_{i=1}^{r(A)}\sigma_i^P(A)=\sum_{i=1}^{n}\sigma_i^P(A), $$ since the singular values $\sigma_{r(A)+1},\ldots,\sigma_n\}$ are zero.