This document defines eigenvalues and eigenvectors, and discusses their properties and applications. It begins by defining eigenvalues as factors by which eigenvectors are scaled when multiplied by a matrix, with eigenvectors being non-zero vectors that remain proportional after multiplication. The document then covers determining eigenvalues and eigenvectors, properties of diagonalizable matrices, and theorems regarding diagonalizability. Examples are provided throughout to illustrate the concepts.

1. EIGENVALUES AND
EIGENVECTORS
MATH 15 - Linear Algebra
Department of Mathematics

2. At the end of this lesson, the students are
expected to :
•Define eigenvalues and eigenvectors
•Prove properties of eigenvalues and eigenvectors
•Determine the eigenvalues and associated eigenvectors
•Determine if a given matrix is diagonalizable
•Identify properties of a diagonalizable matrix
OBJECTIVES

3. DEFINITION:
The prefix eigen- is adopted from the German word
“eigen” for “innate”, “own”.
The eigenvectors are sometimes also called proper
vectors, or characteristic vectors.
Similarly, the eigenvalues are also known as proper
values, or characteristic values.
EIGENVALUE

4. DEFINITION:
The eigenvectors of a square matrix are the non-
zero vectors which, after being multiplied by the
matrix, remain proportional to the original
vector (i.e. change only in magnitude, not in
direction).
For each eigenvector, the corresponding
eigenvalue is the factor by which the eigenvector
changes when multiplied by the matrix.
EIGENVECTORS
A x = l x

5. DEFINITION:
Simple eigenvalues are eigenvalues of A that has
multiplicity of 1,
If the eigenvalue appears as a root more than
once (k>1), then it is an eigenvalue with
multiplicity k.
EIGENVALUES

6. THEOREM:
Since zero vector is a trivial solution satisfying
A x = l x for any scalar , it should not be
considered as an eigenvector.
Proof:
A 0 = l 0
ZERO VECTOR

7. Let . Show that is an
eigenvector with the corresponding
eigenvalue of -2.
EXAMPLE









4
1
16
6
A 





1
2

8. Let
• Find the eigenvalues of A.
• Find the eigenvector corresponding to each
eigenvalue in (a).
EXAMPLES
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
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





1
1
4
1
A

9. • The scalar l is an eigenvalue of a square matrix
A of order n, if and only if
• The equation
is called the characteristic equation of A,
while the expression
is called the characteristic polynomial.
THEOREM
0
)
det( 
 A
In
l
0
)
det( 
 A
In
l
)
det( A
In 
l

10. • The set,
is called the eigenspace of A corresponding to l.
Find the eigenspace of
EIGENSPACE
}
|
|
{ v
Av
V
x
E l
l 



11. If A is a triangular matrix of order n then the
eigenvalues are the diagonal entries.
Find the eigenvalues of matrix A.
THEOREM
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









6
0
0
0
8
4
0
0
5
2
2
0
4
1
0
1
A

12. If A is a square matrix of order n with
eigenvalues l1, l2, l3,..., ln then,
A: det (A) = | A |= l1 (l2) (l3) ...(ln)
B: trace (A) = sum of diagonal entries
trace (A) = l1 + l2 + l3+ ...+ln
Example:
Prove the theorem by finding all the eigenvalues
of A
THEOREM
.

13. DEFINITION:
If in the expression D = P -1AP, there exist a
matrix P and its inverse P -1, such that D will
be a diagonal matrix, then A is diagonalizable
matrix
Two square matrices A and B are similar if there exists
an invertible matrix P such that B = P-1 AP. Matrices
that are similar to diagonal matrices are said to be
diagonalizable.
DIAGONALIZATION

14. DEFINITION:
If in the expression D = P -1AP, there exist a
matrix P and its inverse P -1, such that D will
be a diagonal matrix, then A is diagonalizable
matrix
Two square matrices A and B are similar if there exists
an invertible matrix P such that B = P-1 AP. Matrices
that are similar to diagonal matrices are said to be
diagonalizable.
DIAGONALIZATION

15. Show that P will diagonalize A given that :
A
EXAMPLE

16. DEFINITION:
If D is a diagonal matrix, then clearly it is
diagonalizable because we can find an
invertible matrix P = In which is the identity
matrix such that D = In A I. Thus, every
diagonal matrix is diagonalizable.
Example: Show that A is diagonalizable
THEOREM
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
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
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
6
0
0
0
8
4
0
0
5
2
2
0
4
1
0
1
A

17. DEFINITION:
If D is a diagonal matrix, then clearly it is
diagonalizable because we can find an
invertible matrix P = In which is the identity
matrix such that D = In A I. Thus, every
diagonal matrix is diagonalizable.
Example: Show that A is diagonalizable
THEOREM
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












6
0
0
0
8
4
0
0
5
2
2
0
4
1
0
1
A

18. THEOREM 1:
A square matrix of order n is said to be
diagonalizable if and only if A has n linearly
independent eigenvectors. Moreover, the
diagonal entries of D are the eigenvalues of A
and the columns of P are the corresponding
eigenvectors.
THEOREM 2
Let A be a square matrix of order n. Then A is
diagonalizable if A has n distinct eigenvalues.
THEOREM

19. Which of the matrices A and B are
diagonalizable.
EXAMPLE

20. Find matrix P that will diagonalize matrix A
EXAMPLE

21. Determine whether A is diagonalizable
EXAMPLE

22. Every square matrix with order n that is real
symmetric is diagonalizable.
Show that A is diagonalizable
THEOREM

23. TEXTBOOKS
Elementary Linear Algebra, Bernard Kolman and David
R. Hill, 7th ed., 2003
WEBSITES:
http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_a
nd_eigenspace
SUGGESTED READINGS