The Moore-Penrose Pseudoinverse

Let V and W be inner product spaces finitely generated over the same field of scalars and let α: V → W. We know that there exists a linear transformation α-1: W → V such that α -1 is the identity automorphism on V and αα -1 is the identity automorphism on W if and only if dim(V) = dim(W) and α is an isomorphism. If this is not the case, we would like to weaken the notion of “inverse” so that some similar condition can exist. There are many possibilities of doing this, and we will bring only one of them here. Given a linear transformation α: V→ W as above, we will say that a linear transformation β: W → V is the Moore-Penrose pseudoinverse of α if and only if the following conditions are satisfied: (1) αβα = α; (2) βαβ = β; (3) βα: V → V is self-adjoint; (4) αβ: W → W is self-adjoint.