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Exercise 7.2.17

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1.

Assume that

x = v_{1} + v_{2} + \dots + v_{k}

and

y = u_{1} + u_{2} + \dots + u_{k} .

Then $S$ is a linear operator since

S (x + cy) = λ_{1} (v_{1} + c u_{1}) + \dots + λ_{k} (v_{k} + c u_{k})

= S (x) + cS (y) .

Observe that if $v$ is an element in $K_{λ}$ , then $S (v) = λv$ . This means that if we pick a Jordan canonical basis $β$ of $T$ for $V$ , then ${[S]}_{β}$ is diagonal.

2.

Let

β

be a Jordan canonical basis for

T

. By the previous argument, we have

{[T]}_{β} = J

and

{[S]}_{β} = D

, where

J

is the Jordan canonical form of

T

and

D

is the diagonal matrix given by

S

. Also, by the definition of

S

, we know that

{[U]}_{β} = J - D

is an upper triangular matrix with each diagonal entry equal to zero. By Exercise 7.2.11 and Exercise 7.2.12 the operator

U

is nilpotent. And the fact

U

and

S

commutes is due to the fact

J - D

and

D

commutes. The latter fact comes from some direct computation.

jephianlin

2011-06-27 00:00

Exercise 7.2.17

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