Suppose we have freely chosen two matrices A and B, which are 3$\times$3, real and symmetric i.e.:
$A= \left( \begin{array}{ccc} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{array} \right), $
$B= \left( \begin{array}{ccc} \alpha_1 & 0 & 0 \\ 0 & \alpha_2 & 0 \\ 0 & 0 & \alpha_3 \end{array} \right). $
How we can estimate what is the range of values (min, max) for $\text{Tr}(AB)$? I start from Cauchy-Schwarz inequality, where I land on $-\sqrt{\text{Tr}(AA)\text{Tr}(BB)}\leq\text{Tr}(AB)\leq\sqrt{\text{Tr}(AA)\text{Tr}(BB)}$ and how to move next?
We have $AB= \left( \begin{array}{ccc} \lambda_1 \alpha_1 & 0 & 0 \\ 0 & \lambda_2 \alpha_2& 0 \\ 0 & 0 & \lambda_3\alpha_3 \end{array} \right),$
Hence $Tr(AB)=\lambda_1 \alpha_1+\lambda_2 \alpha_2+\lambda_3 \alpha_3$