Exercise 7.1.9

Answers

1.

The subspace

W

T

-invariant by Exercise 7.1.4. Let

{v_{i}}_{i}

be the cycle with initial vector

v_{1}

. Then

{[T_{W}]}_{γ}

is a Jordan block by the fact

T_{W} (v_{1}) = λ v_{1}

and

T_{W} (v_{i}) = v_{i - 1} + λ v_{i}

for all $i > 1$ . And $β$ is a Jordan canonical basis since each cycle forms a block.

2.

If the

ii

-entry of

{[T]}_{β}

λ

, then

v_{i}

is an nonzero element in

K_{λ}

. Since

K_{λ} \cap K_{μ} = {0}

for distinct eigenvalues

λ

and

μ

, we know that

β^{'}

is exactly those

v_{i}

’s such that the

ii

-entry of

{[T]}_{β}

λ

. Let

m

be the number of the eigenvalue

λ

in the diagonal of

{[T]}_{β}

. Since a Jordan form is upper triangular. We know that

m

is the multiplicity of

λ

. By Theorem 7.4(c) we have

\dim (K_{λ}) = m = | β^{'} | .

So $β^{'}$ is a basis for $K_{λ}$ .

jephianlin

2011-06-27 00:00