Let $T$ be a self-adjoint operator in V and $A=[T]_B$ where $B$ s an orthonormal basis of $V$.
So, I have to prove that $T$ is positive definite if and only if $L_A$ is positive definite.
I haven't tried much since I don't really known where to start. I've proven that $T$ is definite positive if and only if all its eigen values are positive, but I don't know if that will help.
Best Answer
Suppose $A$ is a positive definite operator. Fix an orthonormal basis $\left\{ e_{j}\right\} $ in V. For all $v=\sum v_{j}e_{j}$, $$ \sum_{ij}v_{i}\overline{v_{j}}\left\langle Ae_{i},e_{j}\right\rangle =\left\langle Av,v\right\rangle \geq0. $$ Thus, the matrix of $A$, i.e., $\left(\left\langle Ae_{i},e_{j}\right\rangle \right)_{ij}$, is positive definite.