Moore-Penrose Pseudo-Inversion, part 1

Based on Ian's Introduction to the Moore-Penrose Pseudo-Inversion

Consider the function y, where the operator matrix transforms the function x,

$$y = {\bf{A}}x$$

We multiply both sides by the matrix transpose, A^T,

$${{\bf{A}}^{\rm{T}}}y = {{\bf{A}}^{\rm{T}}}{\bf{A}}x$$

We must be careful when laying out the equation, because the order in which matrices appear, as with operators, is important.

$$\left( {{{\bf{A}}^{\rm{T}}}y} \right) = \left( {{{\bf{A}}^{\rm{T}}}{\bf{A}}} \right)x$$

In this form we see that we can express the left hand side as a function of x, involving the combined matrices A^TA,

$${{\bf{A}}^{\rm{T}}}y = {\bf{B}}x$$

We note that this structure is topologically equivalent to,

$$q = {\bf{B}}x$$

where
$$q = {{\bf{A}}^{\rm{T}}}y$$		$${\bf{B}} = {{\bf{A}}^{\rm{T}}}{\bf{A}}$$

At this point, to find x, we use the inverse of matrix B,

$$x = {{\bf{B}}^{ - 1}}q$$

By this token, since the forms are topologically equivalent, we have,

$$x = {\left( {{{\bf{A}}^{\rm{T}}}{\bf{A}}} \right)^{ - 1}}{{\bf{A}}^{\rm{T}}}y$$