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Math Help - spectral decomposition

  1. #1
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    spectral decomposition

    Consider the spectral decomposition , let
    Gj=xjyjT for j=1,2,3 ,n so that A=segma mjGj .

    a-Show that each Gj has rank one and that G1,..Gn are mutually orthogonal in the sense that GkGj=mkjGk for 1<j , k<n
    b-if p(l) is any scalar polynomial show that p(A) is simple and p(A)= segma p(mj)Gj.
    The matrix Gj, 1<j<n, is referred to as the constituent matrix of A associated with mj .
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    Re: spectral decomposition

    Quote Originally Posted by bernoli View Post
    Consider the spectral decomposition , let
    Gj=xjyjT for j=1,2,3 ,n so that A=segma mjGj .

    a-Show that each Gj has rank one and that G1,..Gn are mutually orthogonal in the sense that GkGj=mkjGk for 1<j , k<n
    b-if p(l) is any scalar polynomial show that p(A) is simple and p(A)= segma p(mj)Gj.
    The matrix Gj, 1<j<n, is referred to as the constituent matrix of A associated with mj .
    What have you been able to do on this problem so far?

    -Dan
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