# Is $\| B(AB)^\dagger \|_2$ uniformly bounded for all positive diagonal matrices $B$?

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 $$0. Then $$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.