I'm trying to show that the adjoint transformation $T^*$ of the endomorphism $T$ on a finite dimensional, real inner product space has the same characteristic polynomial as $T$ in a coordinate free (so no matrices) way.
Definition of adjoint of endomorphism $T$ on the VS $V$: It's the transformation such that $$\langle T(x), y\rangle = \langle x, T^*(y)\rangle$$ for all $x,y\in V$.
It's easy to show that $(\lambda\ id - T)^* = \lambda\ id - T^*$ where $id$ is the identity transformation but I can't figure out how to either get an explicit expression for $T^*$ or in some other way apply a determinant to a transformation in an inner product. Any ideas?