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Let $A$ be a $k\times n, k<n-1$ random matrix with entries drawn i.i.d. from a standard Gaussian, and $B$ a $k\times m$ random matrix with entries drawn i.i.d from a standard Gaussian, and independently of the entries of $A$.

Is there any upper bound known about the largest singular value of this matrix: $(BB^T)^{-1/2}A$?

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    $\begingroup$ The probability that the largest singular value exceeds $T$ should be something like $T^{-(n-k+1)}$. I do some closely related estimates in section 4.2.5 of my paper with Froyland and González-Tokman, arxiv.org/pdf/1310.2398 $\endgroup$ – Anthony Quas Sep 15 '15 at 4:38
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    $\begingroup$ The case $k=m$ is somewhat special, as it leads to the Jacobi ensemble (in particular edge asymptotics are known in great detail). See Forrester's book and also his recent article arxiv.org/pdf/1401.2572v2.pdf $\endgroup$ – ofer zeitouni Sep 17 '15 at 5:13

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