linear algebra

Eigenvalues, Eigenvectors and Diagnolization

August 17, 2020

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.

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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?

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