EIGENVALUES AND EIGENVECTORS
DEFINITION
Let A be an nxn matrix. An eigenvector is A is a nonzero vector x such that A x  λx for some
scalar λ . A scalar λ is called an eigenvalue of A if there is a nontrivial solution x of A x  λx .
Such an x is called an eigenvector corresponding to λ .
EXAMPLE 1
 1 6
 6
 3
 , u  
 , and v  
 . Are u and v eigenvectors of A?
Let A  
5 2
  5
  2
Solution:
 1 6  6    24 
 6

  
  4
  4u
Au  
 5 2   5   20 
  5
So u is an eigenvector of A.
 1 6  3    9 

  
  λv
A v  
5
2

2
11


 

So v is not an eigenvector of A.
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EXAMPLE 2
 1 6
 . Show that 7 is an eigenvalue of A and find the corresponding eigenvectors.
Let A  
5 2
Solution:
7 is an eigenvalue of A iff A x  7x for some nonzero vector x .
A x  7x is equivalent to A x  7x  0 or A  7I 2 x  0 . This last equation is the one we will
work with.
6
 1 6
 1 0   6
  7
  

A  7I 2  
5 2
 0 1  5  5 
Notice that the columns of this matrix are linearly dependent. So A  7I 2 x  0 will have
infinitely many solutions. More specifically, it will have nontrivial solutions so 7 is an eigenvalue
of A.
 6
6 0   1  1 0 

~
 5  5 0 0
0 0

 

1
 1
The solution set to A  7I 2 x  0 is x  x 2   . Each vector of the form x 2   is an
1
 1
eigenvector corresponding to the eigenvalue λ  7 .
DEFINITION
Let A be an nxn matrix. The characteristic equation of A is detA  λIn   0 .
THEOREM
A scalar λ is an eigenvalue of an nxn matrix A if and only if λ is a solution to the characteristic
equation of A; detA  λIn   0 .
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EXAMPLE 3
6  1
5  2


3 8
0
0
Let A  
. Find the eigenvalues of A.
0
0
5
4


0
0
0
1

Solution:
detA  λI n 
5λ 2
6
0
3λ 8

0
0
5λ
0
0
0
1
0
 5  λ 3  λ 5  λ 1  λ   0
4
1 λ
So the eigenvalues of A are 5, 3, and 1.
THEOREM
Let A be an nxn matrix. Then A is invertible if and only if the number 0 is not an eigenvalue of A.
THEOREM
If v 1, v 2 ,  , v k are eigenvectors that correspond to distinct eigenvalues λ1, λ 2 ,  , λ k of
an nxn matrix A, then the set
v,
1

v 2 ,  , v k is linearly independent.
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EXERCISES
For problems 1- 4, determine if λ is an eigenvalue of A.
1.)
3 2

λ  2 and A  
3 8
2.)
3
7

λ  2 and A  
 3  1
3.)
 3 0  1


1
λ  4 and A   2 3
 3 4
5 

4.)
2 2
1


λ  3 and A   3  2 1
0
1 1

For problems 5 - 8, determine if u is an eigenvector of A.
5.)
 1
  3 1

u    and A  
 4
  3 8
6.)
 1  2 
 2 1
 and A  
u
 1 4 


1




7.)
7 9
 4
 3




u    3  and A    4  5 1
 2
 1
4 4 



8.)


u  


1
3 6 7



2  and A   3 3 7 
5 6 5
1


Find the characteristic polynomial and eigenvalues of the matrices in problems 9 - 14.
9.)
2 7

A  
7 2
3  2

10.) A  
 1  1
 2 1

11.) A  
  1 4
 5 3

12.) A  
  4 4
0
0 0


13.) A   0 2
5
 0 0  1


0
2
4  7


0
3

4
6


15.) A  
0
0
3  8


0
0
0
1

0 0
5

8

4 0

16.) A  
0
7 1

1  5 2




17.) A  




3
0
5
1
3
8
0 7
4
1
0

0
0

1
0
0
0
2
9 
0
0
0
1
2
0

0
0

0

3
0
4 0


0
14.) A   0 0
 1 0  3


4
1 2 3 


18.) For A  1 2 3  , find one eigenvalue with no calculation. Justify your answer.
1 2 3 


5 5 5


19.) Without calculation, find one eigenvalue and two linearly independent eigenvectors of A   5 5 5  .
5 5 5


Justify your answer.
20.) Use a property of determinants to show that A and AT have the same characteristic polynomial.
21.) Let A be an nxn matrix. Mark each statement True or False. Justify each answer.
a.) If A x  λ x for some vector x , then λ is an eigenvalue of A.
b.) A matrix A is not invertible if and only if 0 is an eigenvalue.
c.) A number c is an eigenvalue of A if and only if the equation A  cI n x  0 has a nontrivial solution.
d.) To find the eigenvalues of A, you must first reduce A to echelon form.
e.) If A x  λ x for some scalar λ , then x is an eigenvector of A.
f.) If v 1 and v 2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues.
g.) The eigenvalues of a matrix are on its main diagonal.
22.) Explain why a 2x2 matrix can have at most two distinct eigenvalues.
23.) Construct an example of a 2x2 matrix with only one distinct eigenvalue.
24.) Let λ be an eigenvalue of an invertible matrix A. Show that λ1 is an eigenvalue of A 1 . [Hint: Suppose a
nonzero vector x satisfies the equation A x  λ x .]
25.) Show that if A 2 is the zero matrix, then the only eigenvalue of A is 0.
26.) Show that A and AT have the same eigenvalues. [Hint: Find out how A  λI n and A T  λI n are related; then
explain why A  λI n is invertible if and only if A T  λI n is invertible.]
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