Exercise 5.2.12

Answers

1.: Let $E_{λ}$ be the eigenspace of $T$ corresponding to $λ$ and $E_{λ^{- 1}}$ be the eigenspace of $T^{- 1}$ corresponding to $λ^{- 1}$ . We want to prove the two spaces are the same. If $v \in E_{λ}$ , we have $T (v) = λv$ and so $v = λ T^{- 1} (v)$ . This means $T^{- 1} (v) = λ^{- 1} v$ and $v \in E_{λ^{- 1}}$ . Conversely, if $v \in E_{λ^{- 1}}$ , we have $T^{- 1} (v) = λ^{- 1} v$ and so $v = λ^{- 1} T (v)$ . This means $T (v) = λv$ and $v \in E_{λ}$ .
2.: By the result of the previous exercise, if $T$ is diagonalizable and invertible, the basis consisting of eigenvectors of $T$ will also be the basis consisting of eigenvectors of $T^{- 1}$ .

jephianlin

2011-06-27 00:00