I am trying to prove that if $T:V \to V$ is a self-adjoint endomorphism on a finite dimensional real inner product space $V$ with the standard inner product, and each eigenvalue of $T$, $\lambda$ is greater than or equal to $\alpha \in \mathbb{R}$ then
$(T\overrightarrow{x},\overrightarrow{x}) \ge \alpha(\overrightarrow{x},\overrightarrow{x})$
I know from the spectral theorem for self adjoint endomorphisms that the eigenvectors of T form an orthanormal basis for V. But I am not sure how to proceed from here.
Any advice would be much appreciated.
Thanks!