TITLE: Some questions about the pseudoinverse of a matrix
QUESTION [7 upvotes]: For every mxn-matrix A with real entries, there exist a unique nxm-matrix B, also 
 with real entries, such that
$$ABA = A$$
$$BAB = B$$
$$AB = (AB)^T$$
$$BA = (BA)^T$$
B is called the pseudoinverse of A.
There is also a complex version, but I am only interested in the real one.
Now my questions :

If A has rational entries, must B also have rational entries ?
How can I calculate the pseudoinverse of a matrix with PARI/GP ?
Is there a simple method to calculate the pseudoinverse by hand for
small matrices ?
Under which condition are the entries of the pseudoinverse integers ?

I know some special properties, for instance, that for invertible square 
matrices A, the pseudoinverse is simply $A^{-1}$ , or that the pseudoinverse
of a zero-matrix is its transposition, but I have not much experience with
general pseudoinverses.

REPLY [2 votes]: For your first question, by elementary row and column operations, every matrix $A$ over the rationals can be written as a matrix product of the form $PDQ$, where $P$ and $Q$ are products of elementary matrices with rational entries and $D$ is a rectangular diagonal matrix over the rationals. It follows that $A^+ = Q^{-1}D^+P^{-1}$ and you are essentially asking if $D^+$ is rational. The answer should be trivial.
For your third question, if $A$ is at most $3\times3$, you may try the formula
$$
A^+ = \lim_{\delta \searrow 0} (A^\top A + \delta I)^{-1} A^\top
= \lim_{\delta \searrow 0} A^\top (A A^\top + \delta I)^{-1}.
$$
For your last question, I'm not sure if there are any nice and general sufficient conditions.