# Math Help - spectral decomposition

1. ## 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 .

2. ## Re: spectral decomposition

Originally Posted by bernoli
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