Is it true that $A\geq B$ implies $B^{\dagger}\geq A^{\dagger}$ for singular positive semidefinite matrices?

Question

Here $A^{\dagger}$ is the Moore-Penrose pseudo-inverse and $\geq$ denotes the Loewner partial order for positive semidefinite matrices.

I know that the statement is true for non-singular matrices, but cannot find whether this extends to singular matrices, and continuity arguments seem to not apply here.

Any help will be greatly appreciated.

Answer 1

It isn't true. The example could be the following: $$ A=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad B=\begin{pmatrix}1&0\\0&0\end{pmatrix}. $$ Their pseudoinverse matrices are $$ A^+=A,\quad B^+=B. $$ Which implies $$ A\ge B,\quad B^+\not\ge A^+. $$