7.2. Cyclic decomposition and rational forms
7.2. Cyclic decomposition and rational forms
7.2. Cyclic decomposition and rational forms
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• Corollary. T linear operator V.<br />
– (a) There exists a in V s.t. T-annihilator of a<br />
=minpoly T.<br />
– (b) T has a cyclic vector minpoly for T agrees<br />
with charpoly T.<br />
• Proof:<br />
– (a) Let W 0 ={0}. Then V=Z(a 1 ;T) ⊕… ⊕Z(a r ;T).<br />
– Since p i all divides p 1 , p 1 (T)(a i )=0 for all i <strong>and</strong> p 1 (T)=0. p 1 is in<br />
Ann(T).<br />
– p 1 is the minimal degree monic poly killing a 1 . Elements of<br />
Ann(T) also kill a 1 .<br />
– p 1 is the minimal degree monic polynomial of Ann(T).<br />
– p 1 is the minimal polynomial of T.
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