Exercise 7.2.10

Answers

1.: By Theorem 7.4(c), the dimension of $K_{λ}$ is the multiplicity of $λ$ . And the multiplicity of $λ$ is the sum of the lengths of all the blocks corresponding to $λ$ since a Jordan canonical form is always upper triangular.
2.: Since $E_{λ} \subset K_{λ}$ , these two subspaces are the same if and only if they have the same dimension. The previous argument provides the desired result since the dimension of $E_{λ}$ is the number of blocks corresponding to $λ$ . The dimensions of them are the same if and only if all the related blocks have a size $1 \times 1$ .

jephianlin

2011-06-27 00:00