Let $H$ be an finite dim Hilbert space with $|\phi \rangle$ in H, $\langle\phi \mid \phi\rangle=1$, and $\rho \triangleq|\phi\rangle\langle\phi| .$ Let $A$ be an observable which is represented by the Hermitian operator $\hat{A}$ ($\langle A \rangle$ is the avg observation). I've been trying to show the following that: $$1. \langle A\rangle=\operatorname{Tr}(\rho \hat{A}) $$ To prove the above I've been trying to show the eigenvalues of the right hand side are equal to the left, but have been unsuccessful in doing so. Any recommendations would be appreciated $$2. \operatorname{Tr}(\rho)=1$$ I'm not really sure where to start with this question: $$3. \rho^{\dagger}=\rho$$ This would show that $\rho$ is hermitian. I've been trying to show that $p = p^\dagger = G^{-1} p G$ where $G$ is the Gram matrix of the space, but have had issues with simplification.
Thank you for your help!
Hint: The key information that you need for parts 1 and 2 is that for $|\phi\rangle,|\psi\rangle \in H$, we have $$ \operatorname{Tr}(|\psi \rangle \langle \phi|) = \langle \phi|\psi \rangle. $$ This can either be shown directly or as a consequence of the general fact that for operators $A,B,$ $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$.
For part 3, I have no idea what you mean by the "Gram matrix of the space". This fact can be shown as a consequence of the general fact that for operators $A,B,$ $(AB)^\dagger = B^\dagger A^\dagger$.