Let $V$ be a finite dimensional inner product space over $\Bbb R$.
Let $T$ and $U$ be linear self-adjoint operator on $V$.
Assume $U$ is positive definite.
Show all the eigenvalues of $TU$ are real.
Let $\lambda$ be an eigenvalue of $TU$, with corresponding eigenvector $x$. Then
\begin{align}
\left<\lambda x,x \right> &= \left< \left(TU\right)x,x\right> \\
&= \left< x,\left(TU\right)^* x\right> \\
&= \left
I don't see why step (1) holds.
That is, how to show $TU$ is self-adjoint.