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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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