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Exercise 5.4.25

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1.

Let

E_{λ}

be a eigenspace of

T

corresponding to the eigenvalue

λ

. For every

v \in E_{λ}

we have

TU (v) = UT (v) = λU (v) .

This means that $E_{λ}$ is an $U$ -invariant subspace. Applying the previous exercise to each $E_{λ}$ , we may find a basis $β_{λ}$ for $E_{λ}$ such that ${[U_{E_{λ}}]}_{β_{λ}}$ is diagonal. Take $β$ to be the union of each $β_{λ}$ and then both ${[T]}_{β}$ and ${[U]}_{β}$ are diagonal simultaneously.

2.

Let

A

and

B

are two

n \times n

matrices. If

AB = BA

, then

A

and

B

are simultaneously diagonalizable. To prove this we may apply the version of linear transformation to

L_{A}

and

L_{B}

jephianlin

2011-06-27 00:00

Exercise 5.4.25

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