# Does the Moore-Penrose pseudoinverse minimize $\Vert \mathbf{A}^{+}\mathbf{A} - \mathbf{I}_{n}\Vert$?

Does the Moore-Penrose pseudoinverse matrix $$\mathbf{A}^{+}$$ bring the product matrices as close as possible to the relevant identity matrix? Can we say

Given $$\mathbf{A}\in\mathbb{C}^{m\times n}_{\rho}$$, $$\mathbf{A}^{+} = \{\mathbf{A}^{+}\in\mathbb{C}^{n\times m}_{\rho}\colon \Vert \mathbf{A}^{+}\mathbf{A} - \mathbf{I}_{n}\Vert_{2} \wedge \Vert \mathbf{A}\mathbf{A}^{+} - \mathbf{I}_{m}\Vert_{2} \text{ are minimized}\}$$

If so, how would a proof begin?