Given real vector $s=[\sqrt s_1 \quad \sqrt s_2]'$, such that $s_1 \geq s_2$.
Variable is $H\in \mathbb{R}^{2\times 2}$. $H$ is positive semi-definite. I want to know what structure of $H$ will maximize $s'Hs$:
\begin{array}{ll}\max\limits_{{H: H\geq 0}} s'Hs.\end{array}
My attempt: Since I am interested on the structure of $H$, first I normalize it. Let this be normalization of $H$: \begin{equation} \hat{H}= \begin{bmatrix} h_1 & h_2 \\ h_2 & h_3 \end{bmatrix}. \end{equation}
Then rewrite the condition:
\begin{array}{ll}\max\limits_{{\hat{H}: h_1+h_3=1, \\h_2^2\leq h_1h_3}} s'\hat{H}s=\max\limits_{{H: h_1+h_3=1, \\h_2^2\leq h_1h_3}} h_1s_1+2h_2\sqrt{s_1s_2}+h_3s_2=\max\limits_{{h_1: 1>h_1>0}} (\sqrt{h_1s_1}+\sqrt{(1-h_1)s_2})^2.\end{array}
Now I can say only that $h_2=\sqrt{h_1h_3}$ gives maximum, i.e. $\hat{H}$ is rank 1. Can we say more about the structure of $\hat{H}$?