Let $V$ be a finite-dimensional real inner product space and suppose $T$ is an endomorphism on $V$. Show that $(T+T^*)/2$ is self-adjoint and show that there is an orthonormal basis $\{\vec{v_1},...,\vec{v_n}\}$ of V consisting of eigenvectors of $(T+T^*)/2$ such that the eigenvalue corresponding to $\vec{v_i}$ is $(T\vec{v_i},\vec{v_i})$
I know how to prove the first part but I have no idea how to show that such basis exists.