STAT312 Lecture : Spectral decomposition.pdf
5 Apr 2014
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We won t cover the proof of this in class, but here it is if you re interested: de ne an oper- ation a by taking a transpose and a complex conjugate: (a ) = . Note that (ab) = b a and that v v = P| v mv = v v; taking the conjugate transpose of each side gives v mv = v v. V v = 0; so that (why?) is. Eigenvectors corresponding to distinct eigenval- ues are orthogonal. = 1 2 and 1 6= 2 then v01mv2 = v01 (mv2) = 2v01v2 and = v01m v2 = 1v01v2: v01v2 = 0 and so v01v2 = 0. thus ( 1. If is a multiple root of the characteristic equa- tion, with multiplicity. It can be shown that, for a real symmetric ma- trix, this vector space has dimension (this is the only part of this development that requires some work, and is being omitted here).
