For two real symmetric and positive semidefinite matrices, ${\bf A}$ and ${\bf B}$, of same size, can we prove
${\bf A} \succeq {\bf B} \quad \iff \quad {\bf A}^{1/2} \succeq {\bf B}^{1/2}$,
where $\cdot^{1/2}$ denotes the square root of a matrix.
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how do you define the square root for non-SPD matrices? – Exodd Oct 20 '21 at 12:31
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@Exodd, thanks, I have edited the conditions in my question. – H. H. Oct 20 '21 at 12:35
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1It is true that $A \succeq B \implies A^{1/2} \succeq B^{1/2}$, but the converse $A^{1/2} \succeq B^{1/2} \implies A \succeq B$ does not hold. Both of these implications have been asked about on this site; see this post for example. – Ben Grossmann Oct 20 '21 at 13:23