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Exercise 9.A.9

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Proof. Suppose by contradiction that $T \in L (ℝ^{7})$ is such that $T^{2} + T + I$ is nilpotent. Then by Exercise 7 ${(T^{2} + T + I)}_{}$

C $i s a l s o n i l p o t e n t a n d i t s m i n i m a l p o l y n o m i a l i s o f t h e f o r m$ z^j $f o r s o m e p o s i t i v e i n t e g e r$ j $(b e c a u s e$ 0 $i s i t s o n l y e i g e n v a l u e, s e e E x e r c i s e 7 i n s e c t i o n 8 A a n d 8.49) . W e h a v e$

z^{2} + z + 1 = (z - \frac{- 1 + i \sqrt{3}}{2}) (z - \frac{- 1 - i \sqrt{3}}{2}) .

Define $p \in P (ℝ)$ by

p (z) = {(z^{2} + z + 1)}^{j} .

Then $p$ has no real roots and $p (T_{}$

C) = 0 $. T h i s i s a c o n t r a d i c t i o n, b e c a u s e$ p $m u s t b e a p o l y n o m i a l m u l t i p l e o f t h e m i n i m a l p o l y n o m i a l o f$ T_C $(b y 8.46) a n d t h e m i n i m a l p o l y n o m i a l o f$ T_C $h a s a t l e a s t o n e r e a l r o o t (s e e 9.19, 9.11 a n d 8.49) . □$

celiopassos

2017-10-06 00:00

Exercise 9.A.9

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