= Solution
The <spectral theorem for real symmetric matrices> gives $A=Q\Lambda Q^T$ with an <orthogonal matrix> $Q$. <Positive definiteness> makes every <eigenvalue> $\lambda_i$ positive. Define the <matrix square root>
$$
\boxed{A^{1/2}=Q\operatorname{diag}(\sqrt{\lambda_1},\ldots,\sqrt{\lambda_n})Q^T.}
$$
It is <symmetric>, its square is $A$, and for any nonzero <vector> $x$,
$$
x^TA^{1/2}x=\sum_i\sqrt{\lambda_i}\,(Q^Tx)_i^2>0.
$$
Thus it is also <positive-definite>, and its <matrix inverse> is $A^{-1/2}=Q\operatorname{diag}(\lambda_i^{-1/2})Q^T$.
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