Timeline for Almost sure convergence of smallest eigenvector of diagonal matrix

9 events

when toggle format	what		by	license	comment
Sep 12, 2017 at 11:38	vote	accept	PThomasCS
Dec 22, 2015 at 1:26	comment	added	Anthony Quas		I think the idea is that if the spectrum is simple, then the eigenvectors and eigenvalues vary continuously. Hence if the limiting diagonal matrix has almost surely got distinct eigenvalues, then the eigenspaces of the limiting matrix are one-dimensional and are spanned by elements of the standard basis. By continuity, this is also true for nearby matrices.
Dec 21, 2015 at 17:29	answer	added	Liviu Nicolaescu		timeline score: 1
Dec 21, 2015 at 16:42	comment	added	PThomasCS		@LiviuNicolaescu "because in this case, with probability one, the spectrum of $M_n$ is simple." Can you elaborate on why this implies the result?
Dec 21, 2015 at 15:44	comment	added	Liviu Nicolaescu		You need to make some assumptions on the nature of randomness of the matrices $M_n$ that I gather are finite dimensional. If you assume that the entries of $M_n$ are continuous random variables then the result is true because in this case, with probability one, the spectrum of $M_n$ is simple.
Dec 21, 2015 at 14:58	history	edited	PThomasCS	CC BY-SA 3.0	added 28 characters in body
Dec 21, 2015 at 14:51	history	edited	PThomasCS	CC BY-SA 3.0	added 95 characters in body
Dec 21, 2015 at 14:25	review	First posts
Dec 21, 2015 at 14:51
Dec 21, 2015 at 14:21	history	asked	PThomasCS	CC BY-SA 3.0