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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?

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