linear algebra

Eigenvalues, Eigenvectors and Diagnolization

August

1

7,

2

020

Question

1

Find the eigenvalues and eigenvectors of the following matrices.

a

)

(
2 −8
−2 −4

)

b)


2 2 02 0

2

0 2 2




c)



1 0 0 0 0
0 −12 0 0 0
0 0 −12 0 0
0 0 0 92 0
0 0 0 0 −120




Question 2
State whether the following are true or false. If false, explain why or give a
counter-example.

a) Suppose T : R2[x] −→ R2[x] is a linear transformation with eigenvalues
λ1 = 1,λ2 = −2,λ3 = −12. Then T is an isomorphism.
b) A given eigenvector has only 1 eigenvalue associated to it.
c) Suppose A is an n × n matrix, and λ is an eigenvalue for A. Then the
columns of (A−λIn) are linearly independent.
d) A given eigenvalue has only 1 eigenvector associated to it.

1

Question

3

Let B = (1,x,x2) be the standard basis for R2[x], and suppose

T : R2[x] −→ R2[x]

is a linear transformation whose matrix with respect to B is

AT,B =


 5 2 −46 3 −5
10 4 −8




We showed in class that this matrix has the following eigenvectors with as-
sociated eigenvalues;

v1 =


121

2
1


 with λ1 = −1

v2 =


121
1


 with λ2 = 1

v3 =


231

3
1


 with λ3 = 0

a) Show that C = (v1,v2,v3) is a basis for R3.
b) Let S = (e1,e2,e3) be the standard basis for R3. Find

PS−→C (1)
PC−→S (2)

.
c) Find the matrix multiplication

D = (PS−→C)(AT,B)(PC−→S)

d) What is the relationship of this matrix D with respect to the original
transformation T?

2