On the Moore–Penrose inverse from a banal linear algebra viewpoint

[wrobel]

Usually, the theory of the Moore-Penrose inverse looks like some profound material, but it is actually a combination of several standard linear algebra theorems, provided the definition is introduced properly. Such things happen from time to time in peripheral or special fields.
Let $$A:X\to Y,\quad X=\mathbb{R}^m,\quad Y=\mathbb{R}^n$$ be a linear operator and take a subspace $F\subset X$ such that
$$X=F\oplus \ker A.$$
Let $P:X\to F$ be the projection along the subspace $\ker A$.

Theorem 1. There exists a unique operator $B:F\to A(X)$ such that $A=BP.$ The operator $B$ is an isomorphism.

Equip the spaces $X$ and $Y$ with inner products $(\cdot,\cdot)_X$ and $(\cdot,\cdot)_Y$ respectively.
Recall that
$$A^*:Y\to X,\quad (Ax,y)_Y=(x,A^*y)_X.$$

Theorem 2. The following orthogonal decompositions hold:
$$X=\ker A\oplus A^*(Y),\quad \ker A\perp A^*(Y);\quad Y=\ker A^*\oplus A(X),\quad \ker A^*\perp A(X).$$
Let $P_x:X\to A^*(Y),\quad P_y:Y\to A(X)$ be the corresponding orthogonal projections.

From Theorem 1 we know that there is a unique isomorphism $\tilde A: A^*(Y)\to A(X)$ such that $A=\tilde AP_x.$

Definition. We shall call the operator
$$A^+=\tilde A^{-1}P_y:Y\to X$$the MP-inverse operator.

All the standard properties of this operator are easily checked. The main of them are
$$AA^+=P_y,\quad A^+A=P_x.$$
The Hilbert space generalization is evident as well.