Upper bound of the largest eigenvalue of $DAAD$.

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$H = DAAD$, where $D$ is a diagonal matrix and the elements on the diagonal are real numbers (could be positive and negative). $A$ is positive semi-definite and can be diagonalized into $A=U^T\Lambda U$. Then we can convert $H=DU^T\Lambda^2UD$. I am wondering how to estimate the upper bound of the largest eigenvalue of $H$. Please help! Thanks in advance.

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The magic words are "Ostrowski's Theorem":

Ostrowski, A. M., A quantitative formulation of Sylvester’s law of inertia, Proc. Natl. Acad. Sci. USA 45, 740-744 (1959). ZBL0087.01802.