# Largest eigenvalue of matrix product $A^T B A$

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 $$B$$?

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