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I know the eigenvalues and singular values are the same for symmetric, positive definite matrices (e.g., see Why do positive definite symmetric matrices have the same singular values as eigenvalues?). Does this still hold for a symmetric, positive definite random matrices?

Thanks!

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  • $\begingroup$ Why does the random modifer matter? $\endgroup$ – Michael Burr Feb 8 '17 at 0:01
  • $\begingroup$ It holds. The singular values will be random variables as well as the unitary matrix you use for writing down the singular value decomposition. $\endgroup$ – GGG Feb 8 '17 at 3:22
  • $\begingroup$ @MichaelBurr Some properties change when going from deterministic to random and so I'm not sure whether that's my case... $\endgroup$ – CWC Feb 8 '17 at 3:42
  • $\begingroup$ @GGG Thanks. That's what I thought as well, yet did not find any technical documents for that... Do you have any to recommend? $\endgroup$ – CWC Feb 8 '17 at 3:46
  • $\begingroup$ what do you want to know? the distribution of singular values/eigenvalues? My knowlegde on random matrices mostly comes from "An Introduction to Random Matrices" (Anderson, Guionnet, Zeitouni). You may also take a look at Terence Tao's blog. $\endgroup$ – GGG Feb 8 '17 at 4:51

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