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Let the $n \times n$ real matrices ${\bf A}, {\bf B}$ be symmetric and positive semidefinite. If ${\bf A} \succeq {\bf B}$, can one conclude that ${\bf A}^{\frac12} \succeq {\bf B}^{\frac12}$, i.e., does the square root preserve the Löwner order?

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Rodrigo de Azevedo
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  • @user1551 If I may ask, in your opinion, would it be worth generalizing to ${\bf A} \succeq {\bf B} \implies {\bf A}^t \succeq {\bf B}^t$ for all $t \in [0,1]$, or would it be a trivial generalization of this question? – Rodrigo de Azevedo Oct 13 '25 at 22:55
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    This is also a known result, and we say that the function $f:x\mapsto x^r$ is operator monotone on the set of all PSD matrices whenever $r\in[0,1]$. To prove this one may assume that $A\succ0$ first. Then show that the set ${r\in[0,1]: A^r\succeq B^r}$ is midpoint convex. Hence $A^r\succeq B^r$ for all rational $r\in[0,1]$. By continuity the inequality holds for all real $r\in[0,1]$ too. By another continuity argument (or by restricting $A$ and $B$ to the range of $A$), the PSD case follows. – user1551 Oct 13 '25 at 23:37

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