Rank normal form

Any matrix $$\mathbf A$$of rank $r$ can be shown to be equivalent to its rank normal form $$\mathbf N_r$$ as below (see p137 :

$$\mathbf A \sim \mathbf N_r = \begin{pmatrix} \mathbf I_r & \mathbf 0 \\ \mathbf 0 & \mathbf 0 \end{pmatrix}$$