Consider $\| B(AB)^\dagger \|_2$ where $A$ is a real matrix, $B$ is a real, square and symmetric matrix, and $(AB)^\dagger$ is the Moore-Penrose pseudoinverse of $AB$. Is $\| B(AB)^\dagger \|_2$ uniformly bounded over all non-singular symmetric matrices $B$? If not, what about over all positive diagonal matrices $B$?
For example, when $A$ and $B$ are both non-singular, $\| B(AB)^\dagger \|_2 = \| A^{-1} \|_2 $, and the norm of interest is uniformly bounded for all $B$. My question is about whether a similar result exists for when $A$ is not invertible. Some relevant references would be very helpful.
The answer to your first question is "no". Consider e.g. $$ A=\pmatrix{0&1\\ 0&0},\ B=\pmatrix{1&t\\ t&t} $$ where $0<t<1$. Then $$ B(AB)^+ =\pmatrix{1&t\\ t&t}\pmatrix{t&t\\ 0&0}^+ =\pmatrix{1&t\\ t&t}\pmatrix{\frac12t^{-1}&0\\ \frac12t^{-1}&0} =\pmatrix{\frac12(t^{-1}+1)&0\\ 1&0}, $$ whose norm is not bounded above.