Worksheet:  Ch  5  —  Eigenvalues  and  Eigenvectors    

 

   

 

 

   

4. (10 points) Consider the matrix A =

!7 53 !7

"

(a) Find the eigenvalues and eigenvectors of A.

(b) Find matrices P and D such that A = PDP!1 where P is invertible and Ddiagonal.

(c) Compute A3.

7. Let A be a 3 ! 3 matrix with eigenvalues 2, "1 and 3.

(a) (2 Pts.) Find the eigenvalues of A!1.

(b) (2 Pts.) Find the determinant of A.

(c) (2 Pts.) Find the determinant of A!1.

(d) (2 Pts.) Find the eigenvalues of A2 " A.

(a) All eigenvalues are nonzero, then,

Ax = !x ! x = !A!1x, ! A!1x =1

!x.

so the eigenvalues of A!1 are 1/2, "1, 1/3.

(b)

The eigenvalues of A are all di!erent, so A has 3 l.i. eigenvectors and the matrix P constructedwith the eigenvectors as column vectors is invertible. Matrix A is diagonalizable, and A =PDP!1 with D = diag(2,"1, 3).

det(A) = det(PDP!1) = det(P ) det(D)1

det(P )= det(D) = (2)("1)(3) = "6,

so det(A) = "6.

(c)

det(A!1) =1

det(A)= "1

6.

(d) Let x be an eigenvector of A with eigenvalue !, then,

(A2 " A)x = A2x " Ax = !2x" !x = !(! " 1)x,

so the eigenvalues of A2 " A are 2, 2, 6.

5. (a) (4 Pts.) Find all the eigenvalues of the matrix,

A =

!

"3 !1 11 1 !12 !2 2

#

$ .

(b) (1 Pts.) Can you decide how many linearly independent eigenvectors has thismatrix without actually computing them? Justify your answer.

6. Let k be any number, and consider the matrix A given by

A =

!

""#

2 !2 4 !10 3 k 00 0 2 40 0 0 1

$

%%& .

(a) (3 Pts.) Find the eigenvalues of A, and their corresponding multiplicity. Showyour work.

(b) (7 Pts) Find the number k such that there exists an eigenspace EA(!) that istwo dimensional, and find a basis for this EA(!). The notation EA(!) means theeigenspace corresponding to the eigenvalue ! of matrix A. Show your work.

7. (a) (5 Pts.) Find the eigenvalues and eigenvectors of the matrix A =

!2 11 2

". Show

your work.

(b) (3 Pts.) Find matrices P and D such that A = PDP!1, where P is invertible anD diagonal. Show your work.

 

   

   

 

 

   

   

4 MATH. 20F SAMPLE FINAL (WINTER 2010)

(3i) (20 points) Find the eigenvalues and associated eigenvectors of the fol-lowing matrix

A =

!

"

!1 4 !2!3 4 0!3 1 3

#

$ .

The characteristic polynomial of A is equal to

!(! ! 1)(! ! 2)(! ! 3),

so that the eigenvalues are! = 1, 2, 3.

The 1-eigenspace is the nullspace of A!I. A short calculation shows thatthe 1-eigenspace is spanned by the vector (1, 1, 1)T .

Similarly, the 2-eigenspace is spanned by (2, 3, 3)T and the 3-eigenspaceis spanned by (1, 3, 4)T .

MATH. 20F SAMPLE FINAL (WINTER 2010) 5

(ii) (10 points) Find an invertible matrix P and a diagonal matrix D suchthat P!1AP = D.

We take P to be the matrix whose columns are the 3 eigenvectors wefound in (i):

P =

!

"

1 2 11 3 31 3 4

#

$

and D to be the diagonal matrix whose diagonal entries are the correspond-ing eigenvalues:

D =

!

"

12

3

#

$ .

(iii)(5 points) Compute A10. You may express your answer as the productof not more than 3 matrices.

We haveA10 = (PDP!1)10 = PD10P!1.

Moreover, we have

D10 =

!

"

1210

310

#

$

and

P!1 =

!

"

3 !5 3!1 3 !20 !1 1

#

$ .

MATH. 20F SAMPLE FINAL (WINTER 2010) 5

(ii) (10 points) Find an invertible matrix P and a diagonal matrix D suchthat P!1AP = D.

We take P to be the matrix whose columns are the 3 eigenvectors wefound in (i):

P =

!

"

1 2 11 3 31 3 4

#

$

and D to be the diagonal matrix whose diagonal entries are the correspond-ing eigenvalues:

D =

!

"

12

3

#

$ .

(iii)(5 points) Compute A10. You may express your answer as the productof not more than 3 matrices.

We haveA10 = (PDP!1)10 = PD10P!1.

Moreover, we have

D10 =

!

"

1210

310

#

$

and

P!1 =

!

"

3 !5 3!1 3 !20 !1 1

#

$ .

Math 20F Final Exam 11:30 AM March 21, 2003

• There are a total of 80 points possible.• TWO PAGES of notes are allowed. No calculators are allowed.• You must show your work to receive credit.

1. (12 pts) (a) Find the eigenvalues of the matrix

!

"

1 2 00 3 00 0 4

#

$.

(b) Find an eigenvector for each eigenvalue.

2. (12 pts) Let W be the subspace of R5 spanned by

!

%

%

%

"

12020

#

&

&

&

$

,

!

%

%

%

"

2!1002

#

&

&

&

$

and

!

%

%

%

"

020!21

#

&

&

&

$

.

(a) Find an orthonormal basis for W .

(b) Write (9, 9 9, 9, 9)T as a sum of a vector in W and a vector in W!.

3. (6 pts) L((a, b, c)T ) = ax(x! 1) + bx + c defines a linear transformation L from R3 toP3. Find a matrix for L using the standard basis for R3 and the basis 1, x, x2 for P3.

4. (12 pts) A matrix A " R4"4 has eigenvalues 1, !1, 2 and 3. What are the eigenvaluesand determinants of the following matrices?

(i) A#1 (ii) AT (iii) A2 ! A.

5. (10 pts) Let v and w be nonzero vectors in Rn. Define A " Rn"n by A = vwT . Provethat v is a basis for the column space of A.Hint: What is column k of A in terms of v and w?

THERE ARE MORE PROBLEMS

   

   

 

 

 

 

Problem 4. Let A =

1 −2 30 1 00 1 2

.

a) Find all of the eigenvalues of A.

b) Find a basis for the eigenspace corresponding to each of the eigenvalues you found in part(a).

c) Is A diagonalizable? Why or why not?

Problem 4. Let A =

1 −2 30 1 00 1 2

.

a) Find all of the eigenvalues of A.

b) Find a basis for the eigenspace corresponding to each of the eigenvalues you found in part(a).

c) Is A diagonalizable? Why or why not?

Problem 4. Let A =

1 −2 30 1 00 1 2

.

a) Find all of the eigenvalues of A.

b) Find a basis for the eigenspace corresponding to each of the eigenvalues you found in part(a).

c) Is A diagonalizable? Why or why not?