Solving Matrix equation $Y=WX$ for X

Question

I have the following equation of matrices (3 matrices): $Y=WX$

I know that to solve for X we have to do: $W^{-1}Y=W^{-1}WX\xrightarrow{}W^{-1}Y=IX\xrightarrow{}X=W^{-1}Y$

But... can someone let me know how the above equation is equal to: $X=(W^{T}W)^{-1}W^{T}Y$ ?

where $W^{T}$ is the matrix transposed and $W^{-1}$ the inverse of a matrix.

For reference, I found the following equation in page 11 of a PDF in this hyperlink.

Answer 1

We have $$ X=W^{-1}Y=W^{-1}(W^T)^{-1}W^TY=(W^{T}W)^{-1}W^{T}Y. $$

Answer 2

As for all invertible matrices $F$, $G$ we have $(FG)^{-1}=G^{-1}F^{-1}$, it follows that $(W^T W)^{-1}=W^{-1}(W^T)^{-1}$ so $(W^T W)^{-1}W^T = W^{-1}(W^T)^{-1} W^T=W^{-1}$, and so your two expressions are the same for any invertible $W$.

Answer 3

Your first solution

$$Y=WX\implies X=W^{-1}Y$$

is valid only when $W$ is a square and invertible matrix.

But in this case, as indicated in the paper, we are using the least square method and since in general matrix $W$ is not square we can show that the best solution is given by

$$x=(W^{T}W)^{-1}W^{T}y \implies X=(W^{T}W)^{-1}W^{T}Y$$

The matter is related to projection onto subspaces, for a detailed explanation refer to Writing projection in terms of projection matrix.