The question is stated in the title. Where $A$ and $B$ are positive definite real matrices and $\dagger$ stands for the Moore Penrose inverse.
This question link1 seems to be close but it has an aditional assumption that $Ker(A) = Ker(B)$.
2026-03-17 22:59:24.1773788364
Does $ 0 \leq A \leq B \implies 0\leq B^\dagger \leq A^\dagger$?
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This cannot work without the assumption on the null spaces. To see this, take $A=0$, $B\ge0$ but not zero. Then $A^\dagger\le B^\dagger$ but these matrices are not equal, thus the claim is not valid.