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Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix.

What can be said about the distribution of the largest singular value of $PD$?

When $n=1$, I get concentration inequalities from Lemma 1 of this paper. I would like a probabilistic upper bound for the general case, and I wouldn't be surprised if this is already available in the literature.

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I think the following answers your question (and more): http://www-personal.umich.edu/~romanv/papers/product-random-deterministic.pdf

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  • $\begingroup$ As the link of Ofer is now long broken, let me add that he was probably linking to a paper of Roman Vershynin entitled "Spectral norm of products of random and deterministic matrices". It can be found on the web, but I will not offer a link as this is apparently pointless. $\endgroup$ – Fabian Wirth Aug 8 '19 at 16:43

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