Math 121B程序写作、辅导Python，c/c++编程

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Math 121B作业写作、辅导Python，c/c++编程作业、写作Java语言作业
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
anybodys 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 (T) [1/2, 3/4]. (Recall that
(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
SS2if, 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= Sfollows.
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 = AA.
Now, we want to show the other direction, i.e. if AA = AA, 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 = SJS1.