PPT: Eigenvalues and Eigenvectors | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


Eigenvalue Problems
1. Eigenvalues and eigenvectors
2. Vector spaces
3. Linear transformations
4. Matrix diagonalization
Page 2


Eigenvalue Problems
1. Eigenvalues and eigenvectors
2. Vector spaces
3. Linear transformations
4. Matrix diagonalization
The Eigenvalue Problem
 Consider a nxn matrix A
 Vector equation:  Ax = ?x
»
Seek solutions for x and ?
»
 ? satisfying the equation are the eigenvalues
»
Eigenvalues can be real and/or imaginary; distinct and/or 
repeated
»
x satisfying the equation are the eigenvectors
 Nomenclature
»
The set of all eigenvalues is called the spectrum
»
Absolute value of an eigenvalue:
»
The largest of the absolute values of the eigenvalues is 
called the spectral radius
2 2
b a ib a
j j
+ = ? + = ? ?
Page 3


Eigenvalue Problems
1. Eigenvalues and eigenvectors
2. Vector spaces
3. Linear transformations
4. Matrix diagonalization
The Eigenvalue Problem
 Consider a nxn matrix A
 Vector equation:  Ax = ?x
»
Seek solutions for x and ?
»
 ? satisfying the equation are the eigenvalues
»
Eigenvalues can be real and/or imaginary; distinct and/or 
repeated
»
x satisfying the equation are the eigenvectors
 Nomenclature
»
The set of all eigenvalues is called the spectrum
»
Absolute value of an eigenvalue:
»
The largest of the absolute values of the eigenvalues is 
called the spectral radius
2 2
b a ib a
j j
+ = ? + = ? ?
Determining Eigenvalues
 Vector equation
»
Ax = ?x ? (A-??)x = 0
»
A-?? is called the characteristic matrix
 Non-trivial solutions exist if and only if:
»
This is called the characteristic equation
 Characteristic polynomial
»
nth-order polynomial in ?
» Roots are the eigenvalues {?
1
, ?
2
, …, ?
n
}
0 ) det(
2 1
2 22 21
1 12 11
=
- - - = - ?
?
?
?
nn n n
n
n
a a a
a a a
a a a
?
? ? ? ?
?
?
I A
Page 4


Eigenvalue Problems
1. Eigenvalues and eigenvectors
2. Vector spaces
3. Linear transformations
4. Matrix diagonalization
The Eigenvalue Problem
 Consider a nxn matrix A
 Vector equation:  Ax = ?x
»
Seek solutions for x and ?
»
 ? satisfying the equation are the eigenvalues
»
Eigenvalues can be real and/or imaginary; distinct and/or 
repeated
»
x satisfying the equation are the eigenvectors
 Nomenclature
»
The set of all eigenvalues is called the spectrum
»
Absolute value of an eigenvalue:
»
The largest of the absolute values of the eigenvalues is 
called the spectral radius
2 2
b a ib a
j j
+ = ? + = ? ?
Determining Eigenvalues
 Vector equation
»
Ax = ?x ? (A-??)x = 0
»
A-?? is called the characteristic matrix
 Non-trivial solutions exist if and only if:
»
This is called the characteristic equation
 Characteristic polynomial
»
nth-order polynomial in ?
» Roots are the eigenvalues {?
1
, ?
2
, …, ?
n
}
0 ) det(
2 1
2 22 21
1 12 11
=
- - - = - ?
?
?
?
nn n n
n
n
a a a
a a a
a a a
?
? ? ? ?
?
?
I A
Eigenvalue Example
 Characteristic matrix
 Characteristic equation
 Eigenvalues: ?
1
 = -5, ?
2
 = 2
?
?
?
?
?
?
- - - =
?
?
?
?
?
?
- ?
?
?
?
?
?
- = - ?
?
? ?
4 3
2 1
1 0
0 1
4 3
2 1
I A
0 10 3 ) 3 )( 2 ( ) 4 )( 1 (
2
= - + = - - - - = - ? ? ? ? ?I A
Page 5


Eigenvalue Problems
1. Eigenvalues and eigenvectors
2. Vector spaces
3. Linear transformations
4. Matrix diagonalization
The Eigenvalue Problem
 Consider a nxn matrix A
 Vector equation:  Ax = ?x
»
Seek solutions for x and ?
»
 ? satisfying the equation are the eigenvalues
»
Eigenvalues can be real and/or imaginary; distinct and/or 
repeated
»
x satisfying the equation are the eigenvectors
 Nomenclature
»
The set of all eigenvalues is called the spectrum
»
Absolute value of an eigenvalue:
»
The largest of the absolute values of the eigenvalues is 
called the spectral radius
2 2
b a ib a
j j
+ = ? + = ? ?
Determining Eigenvalues
 Vector equation
»
Ax = ?x ? (A-??)x = 0
»
A-?? is called the characteristic matrix
 Non-trivial solutions exist if and only if:
»
This is called the characteristic equation
 Characteristic polynomial
»
nth-order polynomial in ?
» Roots are the eigenvalues {?
1
, ?
2
, …, ?
n
}
0 ) det(
2 1
2 22 21
1 12 11
=
- - - = - ?
?
?
?
nn n n
n
n
a a a
a a a
a a a
?
? ? ? ?
?
?
I A
Eigenvalue Example
 Characteristic matrix
 Characteristic equation
 Eigenvalues: ?
1
 = -5, ?
2
 = 2
?
?
?
?
?
?
- - - =
?
?
?
?
?
?
- ?
?
?
?
?
?
- = - ?
?
? ?
4 3
2 1
1 0
0 1
4 3
2 1
I A
0 10 3 ) 3 )( 2 ( ) 4 )( 1 (
2
= - + = - - - - = - ? ? ? ? ?I A
Eigenvalue Properties
 Eigenvalues of A and A
T
 are equal
 Singular matrix has at least one zero eigenvalue
 Eigenvalues of A
-1
:  1/?
1
, 1/?
2
, …, 1/?
n
 Eigenvalues of diagonal and triangular matrices are 
equal to the diagonal elements
 Trace
 Determinant
?
=
=
n
j
j
Tr
1
) ( ? A
?
=
=
n
j
j
1
? A