Rank

Identities about rank

 * $$\text{rank}(BC) \le \min\{\text{rank}(B), \text{rank}(C)\}$$
 * If $$A=BC$$ with $$B \in \mathbb{m\times r}$$ and $$C \in \mathbb{r\times n}$$, then $$\text{rank}(A) \le r$$. This property is very useful for the application of fast matrix-vector multiplication. I.e., to obtain the product $$Ax$$, we can instead do $$BCx$$ and if $$r$$ is very small compared to $$m$$ or $$n$$, then we'd speed up the computation a lot.