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Math 121B, Final Exam Name (print):

Deadline: 3/20/20, 9pm Student Number:

This exam contains 3 pages (including this cover page) and 4 problems.

Put your name on the top of every page.

Answer all questions, writing in complete sentences as appropriate. The following rules apply:

• Allowable materials.

• Do not write in the table to the right.

• Unless otherwise stated, each answer requires

a justification. Mysterious answers not supported

by mathematical reasoning will not receive

full credit.

• You are allowed to use your books and notes.

You may also look up definitions and information

online. However, you are not allowed to

anybody’s help to answer the questions.

• By signing the following line below, you

declare that your submission is truly your own

work:

Signature:

Problem Points Score

Final - Page 2 of 3 Deadline: 3/20/20, 9pm Name:

1. (15 points) Positive Semidefinite Operators, Spectral Theorem

Recall that for any two selfadjoint operators A, B, the notation A ≥ B indicates that (A − B)

is positive semidefinite. Now, let T be a selfadjoint operator on a finite-dimensional complex

vector space such that 1

a) Use the spectral decomposition of T in order to prove that σ(T) ⊂ [1/2, 3/4]. (Recall that

σ(T) is the set of all eigenvalues of T.)

b) Conclude that T2 ≤ T.

c) Let f ∈ V be an arbitrary vector. Show that limk→∞ kT

k

fk = 0.

2. (25 points) Spectral Theorem

Let V be a finite-dimensional complex inner product space. In this problem, we want to show

that S is selfadjoint if and only if for all f ∈ V we have hSf, fi ∈ R.

a) Prove that if S is selfadjoint, this implies that hSf, fi ∈ R. (This direction was also part of

the midterm and it should be a warm-up exercise for you; the interesting part of the exercise is

to show the other direction.)

b) Now, we want to show the other direction (so, you may not assume that S is selfadjoint).

Show that if ImhSf, fi = 0, then h

S−S∗2if, fi = 0.

c) Define the operator A := 1

2i(S − S∗). Show that A is selfadjoint. Use the result from

part b) to show that if for all f ∈ V we have ImhSf, fi = 0, then hAf, fi = 0 for all f ∈ V .

d) Conclude that A = 0 (here, “0” denotes the zero operator). To this end, use that since

A is selfadjoint, there exists an ONB for V which consists of eigenvectors of A. Conclude that

the only possible possible eigenvalue of A is 0 and thus A = 0, from which S= S∗follows.

3. (25 points) Polar Decomposition

Let A be an invertible linear operator on a finite-dimensional complex vector space V . Recall

that we have shown in class that in this case, there exists a unique unitary operator U such that

A = U|A|. The point of this exercise is to prove the following result: an invertible operator A

is normal if and only if U|A| = |A|U.

a) Show that if U|A| = |A|U, then AA∗ = A∗A.

Now, we want to show the other direction, i.e. if AA∗ = A∗A, then U|A| = |A|U, which is going

to be more difficult.

b) Show that if A is normal, then U|A|

2 = |A|2U.

c) We now want to finish the proof by concluding that U|A|

2 = |A|2U implies U|A| = |A|U.

For notational convenience, define B := |A|

2 and use the spectral theorem to show that there

exists a polynomial g(t) such that g(B) = √

B. Use this to conclude U|A| = |A|U.

Final - Page 3 of 3 Deadline: 3/20/20, 9pm Name:

4. (20 points) Jordan Canonical Form

Let V = C4 and consider the following matrix

(1)

a) Show that the only eigenvalues of A are 4 and 8. What are their multiplicities?

b) Compute the eigenspaces of 4 and 8.

c) What is the Jordan canonical form J of A?

d) Find an invertible matrix S such that A = SJS−1.

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