Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / A / 1 / c / Solution

For a rectangular diagonal matrix $D$, define $D^{+}\in {\mathbb{R}}^{n\times m}$ by transposing its shape and takingEntry by entry, $D{D}^{+}D=D$ and $D^{+}D{D}^{+}=D^{+}$. For $A=U\Sigma V^T$, orthogonality therefore givessince $A{A}^{+}A=U\Sigma {\Sigma }^{+}\Sigma V^T=A$, and similarly $A^{+}A{A}^{+}=A^{+}$. This candidate is the Moore-Penrose inverse. It also makes $A{A}^{+}$ and $A^{+}A$ symmetric, although those two extra conditions are not required by the question's stated definition. The two displayed conditions alone need not specify a unique generalized inverse.