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With $A \in \mathbb{S}^{d \times d}_+$ (symmetric positive semi definite) and $B \in \mathbb{S}^{d \times d}_{++}$ (symmetric positive definite), can we rewrite or upper bound $\lambda_{max}(A^T B A)$ in terms of eigenvalues of $A$ and $B$?

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For a quadratic positive semi-definite matrix $M$ we have $\lambda_\rm{max}(M) = \| M \|_2$, where $\|\cdot \|_2$ is the operator norm corresponding to the euclidean norm. Since operator norms are submultiplicative:

$$\lambda_\rm{max}(A^T B A) = \|A^T B A\|_2 \leq \|A^T\|_2 \|B\|_2 \|A\|_2 = \lambda_\rm{max}(A)^2 \cdot \lambda_\rm{max}(B)$$

Equality can be easily achieved using diagonal matrices.

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