5.2 The Characteristic Equation
To find eigenvalues:
Recall: When we write Ax = λx, the
eigenvalue λ can be 0, but the eigenvector x
corresponding to λ cannot be 0.
We found eigenvectors by solving
(A – λI)x = 0 for x.
Since x cannot be 0, the equation
(A – λI)x = 0 has a non-trivial solution.
This is exactly when (A – λI)x = 0 has a free
variable which happen only when (A – λI) is
not invertible, that is at least one of the
columns of (A – λI) is not a pivot column or
det (A – λI) = 0 (by the IMT).
►The equation det (A – λI) = 0 is called the
characteristic equation.
det (A – λI) is called the characteristic
polynomial.
►Solve det (A – λI) = 0 for λ to find
eigenvalues.
Ex: Find the eigenvalues of
 0 1
A

 6 5 .
Write the characteristic equation:
 0 1   0 
A  I  




6
5
0


 

1 
0  



6
5




Solve the characteristic equation:
det(A – λI) = 0 becomes
0   5      61  0
2  5  6  0
  2  3  0
  2,3
So the eigenvalues are 2 and 3.
Example: Use the characteristic equation to
find eigenvalues of
1 2 1
A  0  5 0
1 8 1
2
1 
1  


A  I   0
5
0 
 1
8
1   
Recall: you use the cofactor expansion to find
the determinant of larger matrices.
Expand along row 2:
1 
1
det  A  I    5   
1 1 


  5   1     1
2
  5     2  1  1
  5     2 
2
2
Set equal to 0 and solve:
det  A  I   0
 5   2  2   0
 5      2  0
  5,0,2
“Is A invertible?”
We can now add two statements to the IMT:
Let A be an nxn matrix, then A is invertible iff:
s. The number 0 is not an eigenvalue of A.
t. det A ≠ 0.
ADD THESE TO THE COPY OF THE
THEOREM YOU ARE MEMORIZING.
Example: Find the eigenvalues of
3 2 3 
A  0 6 10
0 0 2 
We can write the characteristic equation:
2
3 
3  
A  I   0
6   10 
 0
0
2   
det  A  I   0
3   6   2     0
  2,3,6
Could we have saved some time? How?
Definition: The multiplicity of an eigenvalue is
its multiplicity as a root of the characteristic
equation.
Example: Find the characteristic equation of:
2
5
A
9

1
0

0
0

2 5  1
0
0
0 
2  
 5

3


0
0

A  I  
 9
1
3
0 


2
5
1 
 1
0 0
3 0
1 3
det  A  I   0
2   3   3    1     0
2   3   2  1     0
The multiplicities of the eigenvalues are:
Similarity
Note: these matrices have been carefully
chosen to produce nice eigenvalues. In real
life, you would use a computer to find or
approximate eigenvalues. The computers
use the concept of similarity.
Definition: For n x n matrices A and B, we say
A is similar to B if  an invertible matrix P
such that: P-1AP = B or,
PP-1AP = PB
AP P-1 = PB P-1
A = PB P-1
Theorem 5.4: If n x n matrices A and B are
similar, then they have the same
characteristic polynomial and hence the same
eigenvalues.
Proof:
A and B are similar, so  P such that:
B = P-1AP
Write characteristic polynomial for B:
det(B – λI) = det(P-1AP – λI)
= det(P-1AP – P-1λI P)
= det[P-1(A –λI) P]
= (det P-1 )[det(A –λI)](det P)
= det(A –λI)
We will use this in Section 3 to find ways of
writing matrices so that products are easier to
compute.