I'm trying to prove it like any other property of adjoint. So, I need to prove following equality: $(\mathcal{A}^{-1}x, y)=(x, \mathcal({A}^{-1})^*y)$.
I know it's very basic, but how to prove this equality?
Or maybe this property of adjoint is proved in different way?
Thanks.
Best Answer
Hint based on the uniqueness of inverse of a matrix, assuming we work on an inner product linear space $\;V\;$ and, of course, on $\;\langle Au,v\rangle=\langle u, A^*v\rangle\;$ and $\;(A^*)^*=A\;$:
$$\forall\,u,v\in V:\;\langle\, A^*(A^{-1})^*u,v\,\rangle=\langle\,(A^{-1})^*u,Av\,\rangle=\langle\,u,A^{-1}Av\,\rangle=\langle u,v\rangle\implies A^*(A^{-1})^*=I$$
so...