7.2. Cyclic decomposition and rational forms

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• Example 3: T:R 3 ->R 3 linear operator # 5 "6 "6& given by % ( in the standard A = % "1 4 2 ( basis. % 3 "6 "4( – charpolyT=f=(x-1)(x-2) 2 – minpolyT=p=(x-1)(x-2) ! (computed earlier) – Since f=pp 2 , p 2 =(x-2). $ ' – There exists a 1 in V s.t. T-annihilator of a 1 is p and generate a cyclic space of dim 2 and there exists a 2 s.t. T-annihilator of a 2 is (x-2) and has a cyclic space of dim 1.
• The matrix A is similar to B= (using companion matrices) # 0 "2 0& % ( % 1 3 0 ( $ % 0 0 2' ( • Question How to find a 1 and a 2 – In general, almost all vector will be a 1 . (actually choose s.t deg s(a 1 ;W) is maximal.) – Let e 1 =(1,0,0). Then Te ! 1 =(5,-1,3) is not in the span . – Thus, Z(e 1 ;T) has dim 2 ={a(1,0,0)+b(5,-1,3)|a,b in R}={(a+5b,-b,3b)|a,b, in R} ={(x 1 ,x 2 ,x 3 )|x 3 =-3x 2 }. – Z(a 2 :T) is null(T-2I) since p 2 =(x-2) and has dim 1. – Let a 2 =(2,1,0) an eigenvector.
Page 1 and 2: 7.2. Cyclic decomposition and ratio
Page 3 and 4: • Proposition: If W is T-invarian
Page 5 and 6: • Choose b not in W. • T-conduc
Page 7 and 8: Cyclic decomposition theorem • Th
Page 9 and 10: • Corollary. If T is a linear ope
Page 11 and 12: - (b) (->) done before - (
Page 13 and 14: Rational forms • Let B i ={a i ,T
Page 15 and 16: • The char.polyT =char.polyA 1
Page 17: - (ii) minpoly p for T has degree 1
Page 21 and 22: • Let a in V. Then a = b 1 +b 2 +
Page 23 and 24: • Another example T diagonalizabl

7.2. Cyclic decomposition and rational forms

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