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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 $Ker(A) = Ker(B)$.

Manuel
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1 Answers1

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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.

daw
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