PPT: Eigenvalues and Eigenvectors

# 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
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
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
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
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
```

## Engineering Mathematics

65 videos|121 docs|94 tests

## FAQs on PPT: Eigenvalues and Eigenvectors - Engineering Mathematics - Civil Engineering (CE)

 1. What are eigenvalues and eigenvectors?
Ans. Eigenvalues and eigenvectors are concepts in linear algebra that are used to analyze the behavior of linear transformations. Eigenvalues represent the scalar values that scale the eigenvectors when the linear transformation is applied to them.
 2. How are eigenvalues and eigenvectors calculated?
Ans. To calculate the eigenvalues and eigenvectors of a matrix, we need to solve the characteristic equation, which is formed by subtracting a scalar value from the diagonal elements of the matrix and taking its determinant. The eigenvalues are the solutions to this equation, and the corresponding eigenvectors are obtained by solving a system of equations formed by substituting each eigenvalue back into the original matrix equation.
 3. What are the applications of eigenvalues and eigenvectors?
Ans. Eigenvalues and eigenvectors have various applications in different fields. In physics, they are used to study quantum mechanics and determine the energy levels of particles. In computer science, they are employed in data analysis, machine learning, and image processing algorithms. They are also used in structural engineering to analyze the stability of structures and in economics to analyze input-output models.
 4. Can a matrix have more than one eigenvector for the same eigenvalue?
Ans. Yes, a matrix can have multiple eigenvectors corresponding to the same eigenvalue. In fact, eigenvectors associated with the same eigenvalue lie in the same eigenspace. The eigenspace represents all possible vectors that can be scaled by the eigenvalue without changing their direction.
 5. How do eigenvalues and eigenvectors relate to diagonalization of a matrix?
Ans. Diagonalization of a matrix involves finding a diagonal matrix D and an invertible matrix P such that P^-1DP is a diagonal matrix. The diagonal elements of D are the eigenvalues of the original matrix, and the columns of P are the corresponding eigenvectors. Diagonalization is useful in simplifying matrix calculations and solving systems of linear equations.

## Engineering Mathematics

65 videos|121 docs|94 tests

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