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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).

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If a matrix is invertible, then its pseudo-inverse is the same as its inverse.

The formula you're using for the pseudo-inverse, by the way, is only applicable when $A$ has full row rank.

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