Linear Algebra Bits Pilani

1. Chapter 3
Determinants and
Eigenvectors

2. Section 3.4
Eigen Values and Eigenvectors

3. Eigen vectors
A =
X=
Find AX.
Now take X= and find AX








2
1
4
5








1
2








1
4

4. Few things to know about
Eigen values and Eigen vectors
Eigenvalues are often introduced in the
context of linear algebra or matrix theory.
Historically, however, they arose in the study
of quadratic forms and differential
equations.

5. In the 18th century Euler studied the
rotational motion of a rigid body and
discovered the importance of the principal
axes.
Lagrange realized that the principal axes
are the eigenvectors of the inertia matrix.

6. Now it finds applications in
Statistics -Principal component analysis.
Physics- Vibration analysis
Image processing/ Image compression
Population growth models
Stability analysis in control theory
Structural mechanics
Google search engine

7. Definition
If A is an nxn matrix. A real number λ is an
eigen value of A iff there is a non zero n-
vector X such that AX= λX.
This non zero vector X is called an eigen
vector corresponding to the eigen value λ.

8. Problem
If
verify that -1 is an eigen value of A
and X= is the associated Eigen vector.












1
2
2
1
2
1
3
2
2
A










1
0
1

9. Method of finding Eigen Pairs
let X be an eigenvector of A corresponding to the
eigen value λ, then we have AX= λX
This homogeneous system has a nontrivial solution
if …………….(1)
(1) is called the characteristic equation of A.
  0
A I X

  
0
A I

 

10. This is a polynomial equation that has n
roots say
These roots are called the eigen values
of A .
Now, for each λ solve A X= λ X to get
the corresponding eigen vector.
1 2
, ,... n
  

11. Problem
Find the characteristic polynomial of








3
1
1
2
A

12. Problem
Find the eigen values and the associated eigen
vectors of












1
2
2
1
2
1
3
2
2
A

13. Properties of Eigen pairs
 If X is an eigenvector of A associated
with an eigenvalue and k is any nonzero
scalar, kX is also an eigenvector of A
associated with the same eigenvalue.

14. Properties continued
 An eigenvector X of A X= λ X cannot
correspond to more than one eigenvalue of
A.
 If A is an upper ( lower) or diagonal, then
the eigenvalues of A are the elements of the
main diagonal of A.
 A and AT have the same eigenvalues.

15. Properties continued
If λ is an eigenvalue of A with associated
eigenvector X. Then λk is an eigenvalue of Ak
associated with the same eigenvector X.
 If λ is an eigen value of A with associated
eigenvector X. Then 1/ λ is an eigenvalue of
A-1 with associated vector X.
 The sum of the eigenvalues of A = Trace(A).

16. If λ =0 is an eigenvalue of A, then A is not
invertible.
The determinant of A is the product of the
eigenvalues of A.

17. Problem
If A is idempotent, Show that the only
possible eigenvalues of A are 0 and 1.

18. Problems contd..
 If A is a matrix all of whose columns add up
to 1, then 1 is an eigenvalue of A.
If A= , find the eigenvalues
of 











2
0
0
2
3
0
3
2
1
I
A
A
A 2
6
5
3 2
3




19. Find the eigenpairs for the
following matrices













1
2
1
1
0
1
3
4
3
A











0
1
1
1
0
1
1
1
0
A

20. Cayley Hamilton Theorem
Every Square matrix satisfies its
characteristic equation.

21. Problem
Verify Cayley Hamilton theorem for
And hence find A3 and A-1
6 8
4 6
A

 
  

 

22. Applications of Eigenvectors
Google
◦ To rank web pages for a given query, Google
calculates an eigenvector of a matrix A (n n)
◦ = 1 if page i links to page j; 0 otherwise
◦ is the number of links to page j
◦ p is the fraction of pages searched that have
outgoing links (usually taken as 0.85)
1
ij
ij
j n
a
g p
p
c

 
ij
g
j
c


23. Applications of Eigenvectors
(continued)
Google
◦ Let x be an eigenvector corresponding to λ = 1
◦ Normalize x so that the sum of its components
equals 1
◦ This vector gives Google’s PageRank
◦ The determinant method can be used to find
eigenvectors of small matrices, but it is not
practical for large matrices such as A (2.7 billion
2.7 billion in 2002) – Google’s method is
unknown
