Let $A \in M_n(\mathbb{C})$ and $x$ a complex unit vector. Show \begin{equation}(x^*Ax)(x^*A^*x) \leq x^*AA^*x. \;\;[1]\end{equation}
Solution attempt: We know that $AA^*$ is positive semi-definite so it is Hermitian and its eigenvalues are nonnegative. Thus, it has a basis $v_1, \ldots, v_n$ of eigenvectors respectively associated with the nonnegative eigenvalues $\lambda_1,\ldots,\lambda_{n-1}.$
Thus, if $x= c_1 v_1 + \ldots + c_nv_n$, the right hand side of [1] becomes $$\lambda_1 |c_1|^2 +\ldots+\lambda_n|c_n|^2.$$
I no longer know what to do with the left hand side of [1].