 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 rootIt is symmetric, its square is $A$, and for any nonzero vector $x$,Thus it is also positive-definite, and its matrix inverse is $A^{-1/2}=Q\mathrm{diag}({\lambda }_i^{-1/2})Q^T$.