What are the conditions under which a the pseudo-inverse of a matrix is not equal to its inverse?
I have a matrix equation:
$ AXB = C $
which according to Laub (13.14, 13.15) has a solution if
$ AA^+CB^+B = C $
where $A^+$ is the pseudo inverse.
I want to be able to say that there is no solution except when $rank(B) = rank(C)$, and therefore it is appropriate to use a minimization solution since that never happens in practice.
The formula for the psuedo-inverse I am using is from Petersen and Pedersen:
$ A^+ = A^T(AA^T)^{-1} $
The bigger picture is that I am fitting a transition matrix to data using a rather obscure algorithm called "Woods Method", found on page 144-145 in Hal Caswell's matrix population models book, and I am trying to really dial in the math (which is pretty sketchy).