Assignment 6 Stat24300-307500(F14) Due Tuesday, November 18, 2014

Stat24300-307500(F14)
Assignment 6 (three pages)
Due Tuesday, November 18, 2014
Suggested reading: Section 5.1, 5.2, 5.3.
Recommended reading for Stat 30750: Watkins Chapter 5.
Problem assignments:
1. Section 5.2, Exercise 8 on page 250.
2. Section 5.2, Exercise 36 on page 253.
3. Section 5.3, Exercise 10 on page 263.
4. (Simple dynamic system of ordinary differential equations) Similar
to Exercises
1, 2 and 4 inSection 5.1,
0
−0.2
0.3
we want to solve du/dt = Bu with initial condition u(0) =
, where B =
.
5
0.2 −0.3
(a) Here the function u(t) = [v(t) w(t)]T is a vector of two components. Write the given vector equation
du/dt = Bu as two scalar equations.
(b) First find two pure exponential solutions of the form ui (t) = xi eλi t , i = 1, 2, where λi are eigenvalues
of P with eigenvectors xi = [yi zi ]T . Start from the characteristic equation of B and show your steps.
(c) Write the complete solution form u(t) = c1 u1 (t) + c2 u2 (t). Use the given initial value u(0) = [0, 5]T
to find the constant c1 and c2 .
(d) On the v-w plane, plot the trajectory of u(t) = [v(t) w(t)]T at t = log(j 2 ) for j = 2, 3, 4, 5, 10, plus
the given initial u(0) = [0, 5]T . What is the limiting point of the trajectory as t → ∞? (Here log = loge )
(e) What is the trajectory of u(t) if the initial value is u(0) = [3, 2]T ?
(f) Find the relationship between matrix B here and matrix A in Question 3 in this assignment. Explain
the reason for the relationship you found.
5. (Block matrix operation) A, B, C, D are matrices with suitable dimensions. In this exercise, you may use
A 0
A B
= det(A) det(D)
det
= det
0 D
C D
(a) The n × n matrices A, B and A + B are invertible.
i. Show that C = A−1 + B −1 is invertible. (Hint: First find the relation of A−1 + B −1 and A−1 (B + A)B −1 .)
ii. Find an expression for C −1 (with reasons).
iii. Express det(C −1 ) in terms of determinants of A, B and A + B.
A B
(b) Matrix A is n × n and invertible. The block matrix M =
is (n + r) × (n + r).
C D
i. What are the dimensions (numbers of rows and columns) of B, C and D?
ii. If we do one step ‘elimination’, we can fine ‘LU’ decomposition of the block matrix
A B
I
0
A B
M=
=
C D
CA−1 I
0 ∗
Find the expression of the matrix *, provide its dimensions.
iii. Then derive the formula det(M ) = det(A) det(∗). Write out * and show your steps.
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(c) Provide a numerical example of the block matrix M in (b), where A, B, C, D are 2 × 2 matrices such
that AC 6= CA. Show with your example that det(M ) = det(AD −CB) is generally not sure. (When
is it true?)
A B
−1
−1
(d) Derive U and det(U ) for the block matrix U =
when A−1 and D−1 exist.
0 D
A 0
−1
−1
(e) Derive L and det(L ) for the block matrix L =
when A−1 and D−1 exist.
C D
6. State whether the statements below are True or False. Give a reason if your answer is True or a
counterexample if your answer is False.
(a) If A2 = 0nn (zero matrix), then 0 is the only possible eigenvalue of A.
(b) Rank one matrix has only one eigenvalue.
(c) If A is an n × n matrix with rank less than n, then 0 is an eigenvalue of A.
(d) Let A be an n×n matrix, and let I be the n×n identity matrix. If the real number λ is an eigenvalue
of A, then λ3 + 3λ2 − 2λ + 4 is an eigenvalue of A3 + 3A2 − 2A + 4I.
(e) Let A be an n×n matrix and B be an invertible n×n matrix. If B −1 AB = C and x is an eigenvector
of A with eigenvalue λ, then B −1 x is an eigenvector of C with eigenvalue λ.
(f) Let A be an n×n matrix and B be an invertible n×n matrix. If B −1 AB = C and x is an eigenvector
of A with eigenvalue λ, then Bx is an eigenvector of C with eigenvalue λ.
7. Use matrix properties to answer the following questions. Only moderate calculation should be involved. (If
you had to do a lot of computation to answer these questions, you should think hard about why it was unnecessary
to do so.)


1
2 −4
1 
2
4 −8  is a projection matrix.
(a) The matrix P =
21
−4 −8 16
i. What subspace does P project onto?
ii. What is the distance from that subspace to the point (1, 1, 1)?
iii. What are the eigenvalues of P ?

(b) The matrix

0 1 0
A =  0 0 −1  is a rotation matrix.
−1 0 0
i. What is the characteristic polynomial of A?
ii. Factor the characteristic polynomial as (λ − r)(λ − z)(λ − z), where z is a complex number.
What are the two possibilities for z?


0.5 0.3 0.3
(c) The matrix M =  0.1 0.5 0.5  is a Markov matrix. Find the eigenvalues of M
0.4 0.2 0.2
without computing det(M − λI). You need to show how you find each eigenvalues of M .
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8. Suppose that v1 , v2 , · · · , vn are nonzero eigenvectors of an n × n matrix A correspoinding to distinct,
nonzero eigenvalues λ1 , . . . , λn . (So Avj = λj vj 6= 0, λi 6= λj if i 6= j.) Prove that v1 , v2 , · · · , vn are
linearly independent. (Hint: Use mathematical induction on n.)
9. (Required for 30750. Optional for 24300.)
(An important matrix, which you should have created a similar one in the sample midterm)
Find the determinant of the matrix shown below (conjecture a formula by looking at the 2 × 2, 3 × 3
and 4 × 4 cases, then use induction to prove the formula you conjectured) and state why the matrix is
invertible if the real numbers c1 , c2 , c3 , . . . , cn are distinct.


1
1
1
···
1
 c1
c2
c3
···
cn 


2 
2
2
 c1 2
·
·
·
c
c
c
n
3
2




..
..
..
..
..


.
.
.
.
.
n−1
n−1
n−1
n−1
· · · cn
c3
c2
c1
(Hint: During mathematical induction, consider (row(i + 1)) − c1 (row i) starting from the bottom row.)
The following are for 30750 only.
1 0
. First verify that the
10. This exercise investigates perturbation effects on eigenvalues of I =
0 1
characteristic equation is λ2 − 2λ + 1 = 0 with roots (eigenvalues) λ1 = 1 and λ2 = 1.
1− 0
, where 0 < << 1. Find the
0
1
characteristic equation for A. Solve the equation to find eigenvalues λ∗1 and λ∗2 of A.
(a) Perturbation effects on a matrix entry.
Consider A =
(b) Perturbation effects on a coefficient of the characteristic polynomial. Assume that a coefficient is
slightly perturbed during the process of finding the roots of the characteristic equation of I. Solve
the perturbed equation
λ2 − 2λ + (1 − ) = 0
ˆ 1 and λ
ˆ2.
to find the roots λ
(c) Derive the relative errors of the perturbation effects
|λ∗1 − λ1 |
,
|λ∗2 − λ2 |
and
ˆ 1 − λ1 |
|λ
,
ˆ 2 − λ2 |
|λ
.
ˆ i − λi |
|λ
?
(d) Compare and commend on the difference in the effects of perturbation on the matrix entry and on
the coefficient of the characteristic equation (during the process of finding the roots).
In particular, for a tiny perturbation = 10−12 , what is the size of
(e) On the same plot, sketch the graphs of the original characteristic polynomial pI (λ) = λ2 − 2λ + 1
and the perturbed polynomials p (λ) = λ2 − 2λ + (1 − ) over the interval λ ∈ [0.5, 1.5] for each value
of = 0.05, 0.1, 0.2, 0.4. Add a light horizontal line (y = 0) to indicate the locations of the roots.
(f) Does the graphs give some indication of why the roots are so sensitive to the perturbation? Or, what
kind of roots are likely to be sensitive to perturbations? Comment briefly.
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