Exercise 7.2.7

Answers

1.

Let

γ^{'}

be the set

{v_{i}}_{i} = 1^{m}

. The desired result comes from the fact

T (v_{i}) = λ v_{i} + v_{i + 1}

for $1 \leq i \leq m - 1$ and

T (v_{m}) = λ v_{m} .

2.

Let

β

be the standard basis for

𝔽^{n}

and

β^{'}

be the basis obtained from

β

by reversing the order of the vectors in each cycle in

β

. Then we have

{[L_{J}]}_{β} = J

and

{[L_{J}]}_{β^{'}} = J^{t}

. So

J

and

J^{t}

are similar.

3.

Since

J

is the Jordan canonical form of

A

, the two matrices

J

and

A

are similar. By the previous argument,

J

and

J^{t}

are similar. Finally, that

A

and

J

are similar implies that

A^{t}

and

J^{t}

are similar. Hence

A

and

A^{t}

are similar.

jephianlin

2011-06-27 00:00