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Exercise 5.4.26

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Let ${v_{1}, v_{2}, \dots, v_{n}}$ be those eigenvectors corresponding to $n$ distinct eigenvalues. Pick

v = v_{1} + v_{2} + \dots + v_{n}

and say $W$ to be the $T$ -cyclic subspace generated by $v$ . By Exercise 5.4.23 we know that $W$ must contains all $v_{i}$ ’s. But this means the dimension of $W$ is $n$ and the set

{v, T (v), \dots, T^{n - 1} (v)}

is a basis by Theorem 5.22

jephianlin

2011-06-27 00:00

Exercise 5.4.26

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