Review of Eigenvectors and Eigenvalues - Class Notes, Civil Engineering, Semester Notes | EduRev

Created by: Babit Mahajan

: Review of Eigenvectors and Eigenvalues - Class Notes, Civil Engineering, Semester Notes | EduRev

 Page 1


Review of Eigenvectors and 
Eigenvalues
from CliffsNotes Online
http://www.cliffsnotes.com/study_guide/Deter
mining-the-Eigenvectors-of-a-
Matrix.topicArticleId-20807,articleId-
20804.html
1
Page 2


Review of Eigenvectors and 
Eigenvalues
from CliffsNotes Online
http://www.cliffsnotes.com/study_guide/Deter
mining-the-Eigenvectors-of-a-
Matrix.topicArticleId-20807,articleId-
20804.html
1
Definition
The eigenvectors x and eigenvalues  ? of a matrix A satisfy
Ax = ?x
If A is an n x n matrix, then x is an n x 1 vector, and ? is a constant.
The equation can be rewritten as  (A - ?I) x = 0, where I is the 
n x n identity matrix.
2
Page 3


Review of Eigenvectors and 
Eigenvalues
from CliffsNotes Online
http://www.cliffsnotes.com/study_guide/Deter
mining-the-Eigenvectors-of-a-
Matrix.topicArticleId-20807,articleId-
20804.html
1
Definition
The eigenvectors x and eigenvalues  ? of a matrix A satisfy
Ax = ?x
If A is an n x n matrix, then x is an n x 1 vector, and ? is a constant.
The equation can be rewritten as  (A - ?I) x = 0, where I is the 
n x n identity matrix.
2
Computing Eigenvalues
Since x is required to be nonzero, the eigenvalues must satisfy
det(A - ?I) = 0
which is called the characteristic equation. Solving it for values 
of ? gives the eigenvalues of matrix A.
3
Page 4


Review of Eigenvectors and 
Eigenvalues
from CliffsNotes Online
http://www.cliffsnotes.com/study_guide/Deter
mining-the-Eigenvectors-of-a-
Matrix.topicArticleId-20807,articleId-
20804.html
1
Definition
The eigenvectors x and eigenvalues  ? of a matrix A satisfy
Ax = ?x
If A is an n x n matrix, then x is an n x 1 vector, and ? is a constant.
The equation can be rewritten as  (A - ?I) x = 0, where I is the 
n x n identity matrix.
2
Computing Eigenvalues
Since x is required to be nonzero, the eigenvalues must satisfy
det(A - ?I) = 0
which is called the characteristic equation. Solving it for values 
of ? gives the eigenvalues of matrix A.
3
2 X 2 Example 
A =                                  so A - ?I = 
1  -2                                            1 - ? -2
3  -4                                                3     -4 - ?
det(A - ?I)  =  (1 - ?)(-4 - ?) – (3)(-2)
= ?
2
+ 3 ? + 2
Set  ?
2
+ 3 ? + 2 to 0
Then = ? = (-3  +/- sqrt(9-8))/2
So the two values of  ? are  -1  and -2.
4
Page 5


Review of Eigenvectors and 
Eigenvalues
from CliffsNotes Online
http://www.cliffsnotes.com/study_guide/Deter
mining-the-Eigenvectors-of-a-
Matrix.topicArticleId-20807,articleId-
20804.html
1
Definition
The eigenvectors x and eigenvalues  ? of a matrix A satisfy
Ax = ?x
If A is an n x n matrix, then x is an n x 1 vector, and ? is a constant.
The equation can be rewritten as  (A - ?I) x = 0, where I is the 
n x n identity matrix.
2
Computing Eigenvalues
Since x is required to be nonzero, the eigenvalues must satisfy
det(A - ?I) = 0
which is called the characteristic equation. Solving it for values 
of ? gives the eigenvalues of matrix A.
3
2 X 2 Example 
A =                                  so A - ?I = 
1  -2                                            1 - ? -2
3  -4                                                3     -4 - ?
det(A - ?I)  =  (1 - ?)(-4 - ?) – (3)(-2)
= ?
2
+ 3 ? + 2
Set  ?
2
+ 3 ? + 2 to 0
Then = ? = (-3  +/- sqrt(9-8))/2
So the two values of  ? are  -1  and -2.
4
Finding the Eigenvectors
Once you have the eigenvalues, you can plug
them into the equation Ax = ?x to find the
corresponding sets of eigenvectors x. 
1 -2        x
1
=       -1 x
1      so
3 -4        x
2
x
2
x
1
– 2x
2
= -x
1
3x
1
– 4x
2
= -x
2
(1) 2x
1
– 2x
2
= 0
(2) 3x
1
– 3x
2
= 0
These equations are not
independent. If you multiply
(2) by 2/3, you get (1).
The simplest form of (1) and (2) is  x
1
- x
2
= 0, or just x
1
= x
2
.
5
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