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250A Homework 1

Solution by Jaejeong Lee

Question 1 Let

be a linear operator. Show that

has 1 and 2

dimensional invariant subspaces.

Solution The characteristic polynomial of

is real cubic, so it has at least one

real root and an eigenvector

such that

. Note that

is a 1-dimensional invariant subspace of

and

. Now if

,

then

and any 2-dimensional subspace of

is invariant under

. If

, then

is a 2-dimensional invariant subspace, since

for

we have

and

hence

. If

, then

is a 2-dimensional invariant subspace since

. Therefore, in any case,

has a 2-dimensional invariant subspace,

too.

Question 2Let be the space of matrices. What is

? Now define the

linear operator

by

Compute

.

SolutionThe standard basis for is

, where

is a matrix

unit with its only nonzero entry being 1 at

. Thus

. We compute

so the eigenvectors of

are

and the corresponding eigenval-

ues are

. Therefore,

.

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Question 3Van der Monde Determinant. Let be the

matrix with entries

Show that

.

SolutionNote that

... ...

... . . .

...

We use induction on . When

, we verify

Assume now the assertion is true up to

and let

... ...

... . . .

...

Since

is a polynomial of degree

and

for

, we

have

... ...

. . . ...

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by the induction hypothesis. Therefore, we finally get

... ...

... . . .

...

and the assertion is true for n.

Question 4(Anti)commutators.Let

be a finite dimensional vector space. Show

that the mapping

where

obeys the Leibnitz rule

. In addition, verify the

Jacobi identity

Generalize the above laws to the mapping

. Include also new rules which mix both operations.

SolutionWe want to find a kind of the Leibnitz rule

in cases

or

and

,

, or

. For

example, when

and

, we have

(1)

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and when

and

, we have

(2)

Now I present a trickto find such rules. (It may not be a trick but a principle.

Because I dont know why it works, to me, it is a trick.)

Matrices can be assigned two attributes, even and odd. If is assigned odd,

we mark it

, and if even, leave it as it is.

Step1Write down (1) or (2).

e.g.

Step2Assign attributes to

arbitrarily.

e.g.

Step3Regard (odd,odd)-pair and (even,even)-pair as even and other pairs as

odd.

e.g. is odd, so is even. is even and is odd.

Step4If you fi nd

odd

odd

, replace

by

.

e.g.

Step5If an odd, as an operator, goes past another odd, then replace

by

.

e.g.

(

goes past

)

Step6Remove markings

and declare you found a rule.

e.g.

Example1.

Step1

.

Step2

Step3Observe

and

.

Step4

Step5

Step6

Example2.

Step1

.

Step2

Step3Observe

,

, and

.

Step4

Step5

Step6

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Example3.

Step1 .

Step2

Step3

is even,

and

are odd.

Step4

Step5

(No change)

Step6

Find more on your own. Note that you never get

because ofStep3

andStep4.

Question 5 Baker Campbell Hausdorff Formula. Let

be a finite dimensional

vector space and

. Show that

if

. Hint: Develop and solve a differential equa-

tion for

.

SolutionWe need the identity

(proved below) to see

the last equality coming from the commutativity assumptions. Since

we

get the unique solution

(Verify this. We need the commutativity assumptions again. To solve

we observe

.) Plugging we get the

desired equality. To show

, let

and

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compute the Taylor expansion for

;

and so on. Therefore

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