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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?

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  • $\begingroup$ Isn't it $α_1λ_1+α_2λ_2+α_3λ_3$? $\endgroup$ – Jimmy R. Dec 9 '16 at 10:21
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    $\begingroup$ "freely chosen" $3 \times 3$, real and symmetric matrices need not be diagonal. $\endgroup$ – Shraddheya Shendre Dec 9 '16 at 10:21
  • $\begingroup$ I'm not sure where you want to "move next". Both inequalities can be attained for certain concrete $A,B$. $\endgroup$ – Martin Argerami Dec 11 '16 at 18:24
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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 $Tr(AB)=\lambda_1 \alpha_1+\lambda_2 \alpha_2+\lambda_3 \alpha_3$

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