7.2. Cyclic decomposition and rational forms

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Examples • Example 2: V 2-dim.v.s. over F. T:V->V linear operator. The possible cyclic subspace decompositions: – Case (i) minpoly p for T has degree 2. • Minpoly p=charpoly f and T has a cyclic vector. • If p=x 2 +c 1 x+c 0 . Then the companion matrix is of the form: # 0 "c & 0 % $ 1 "c 1 ( '
– (ii) minpoly p for T has degree 1. i.e., T=cI. for c a constant. – Then there exists a1 and a2 in V s.t. V=Z(a 1 ;T)⊕Z(a 2 ;T). 1-dimensional spaces. – p 1 , p 2 T-annihilators of a 1 and a 2 of degree 1. – Since p 2 divides the minimal poly p 1 =(x-c), p 2 =x-c also. – This is a diagonalizable case. " c 0% $ # 0 ' c&
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: • The char.polyT =char.polyA 1
Page 19 and 20: • The matrix A is similar to B= (
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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