SVD, matrix inverse, and pseudoinverse

1. Overview

2. Description

2.1 Inverse Full rank square matrix A

Now I'm going to invert A which is fine we assume for the moment that A is an invertible matrix. So it's square and full rank. And of course, whatever operation you perform on one side of the equation must be repeated on the other side of the equation. So we apply the inverse to the right-hand side as well. Now we know that each of these matrices is individually invertible U and V are orthogonal matrices. So they are definitely invertible and sigma. This singular matrix is a diagonal matrix and every diagonal element contains a non zero value. And that's because we assume that A is full rank.

If you have some matrix that's a diagonal matrix the inverse of that diagonal matrix is simply each individual diagonal element inverted or reciprocated.

And by the way if this were a reduced rank matrix so imagine C were 0 then here you would be trying to compute 1 over 0 which is not defined. That's illegal in mathematics and this is another explanation for why a singular matrix does not have an inverse because you would be trying to compute one over zero.

2.2 Pseudoinverse 

3. Reference