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How can we prove that if $A$ is an $n$ by $n$ Hermitian and invertible matrix, then for some invertible matrix $J$: $J^*AJ=A^{-1}$?

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What have you tried up until now? The spectral theorem looks helpful here.. – Alex Youcis Apr 21 '12 at 18:20
up vote 3 down vote accepted

The simplest choice is probably to take $J=A^{-1}$.

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