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Linear Algebra Homework: Fall 2006, Set 2, Assignments of Linear Algebra

Rochester Institute of Technology (RIT)Linear Algebra

A linear algebra homework assignment from the university of california, berkeley, fall 2006. It includes five problems involving matrix operations, eigenvalues, and eigenvectors. Students are required to find the eigenvalues and eigenvectors of given matrices, diagonalize certain matrices, and apply matrix operations. Problem 48 includes a bonus question that checks for the diagonalizability condition.

Typology: Assignments

2009/2010

Uploaded on 03/28/2010

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Linear Algebra - 1016 - 432
Fall 2006
Set 2
Due: Friday, September 29, by 3 pm, at my office
Please write up the following problems
a. section 4.4 , problems 38 , 42 , 48 ( Also, for problem 48, show that the result fails if A is not
diagonalizable)
b. Let
€
A=
1 2 3
1 2 3
1 2 3
. Write A as
€
vwT
, where v and w are
€
3×1
vectors. Use your result to find a simple
formula for
€
An
, the eigenvalues of A, and a basis for each eigenspace. Is A diagonalizable?
c.
€
A=1
2
3−1
2 0
. Diagonalize A, then use your result to find
€
lim
n→∞ An
.
d. Consider
€
A=1 0
1 2
over
€
Z6
. Find the eigenvalues of A. For each eigenvalue, find an associated
eigenvector. Be careful. You’re not in Kansas anymore, but in
€
Z6
.
e. A recurrence relation is given by
,
€
a1=5
, and
€
an=an−1+2a n−2
, for
€
n≥2
. By diagonalizing the
matrix associated with this recurrence, find a simple formula for
€
an
.

Partial preview of the text

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Linear Algebra - 1016 - 432 Fall 2006 Set 2 Due: Friday, September 29, by 3 pm, at my office Please write up the following problems a. section 4.4 , problems 38 , 42 , 48 ( Also, for problem 48, show that the result fails if A is not diagonalizable) b. Let €

A =

. Write A as € vw T , where v and w are € 3 × 1 vectors. Use your result to find a simple formula for €

A

n , the eigenvalues of A, and a basis for each eigenspace. Is A diagonalizable? c. €

A =

. Diagonalize A, then use your result to find € lim n→∞

A

n . d. Consider €

A =

 over € Z 6. Find the eigenvalues of A. For each eigenvalue, find an associated eigenvector. Be careful. You’re not in Kansas anymore, but in €

Z 6.

e. A recurrence relation is given by € a 0 = 4 , € a 1 = 5 , and € an = an− 1 + 2 an− 2 , for € n ≥ 2. By diagonalizing the matrix associated with this recurrence, find a simple formula for € an.