I am considering such a matrix product, namely $ADA^T$, where $A$ is an $n\times m$ matrix with $n>m$, $D$ is an $m\times m$ positive diagonal matrix. I understand the matrix product is positive semi-definite, but is there an estimation of its largest eigenvalue, saying some relationship with eigenvalues of $D$ or $AA^T$ (or $A^TA$)?
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1We have $\lambda_{\max}(ADA^T) \leq \lambda_{\max}(A^TA)\cdot \lambda_{\max}(D)$. Is that a good enough bound? – Ben Grossmann Apr 19 '20 at 14:41
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Yes, it's desired, but do you have a proof? – Leon Apr 19 '20 at 14:48
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Let $\|A\|$ denote the spectral norm (AKA the spectral norm/ the maximum singular value) of $A$. We have $\lambda_{\max}(ADA^T) \leq \lambda_{\max}(A^TA)\cdot \lambda_{\max}(D)$, since $$ \lambda_{\max}(ADA^T) = \|D^{1/2}A\|^2 \leq (\|D^{1/2}\| \cdot \|A\|)^2 = \|A\|^2 \cdot \|D^{1/2}\|^2 = \lambda_{\max}(A^TA)\cdot \lambda_{\max}(D). $$
Ben Grossmann
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@Omnomnomnom That question was more for the OP. In the affirmative case this answer his/her question and he/she should accept the answer :) – lcv Apr 19 '20 at 18:52