Eigenvalues and Eigenvectors
-
Upload
vinod-srivastava -
Category
Engineering
-
view
46 -
download
1
Transcript of Eigenvalues and Eigenvectors
SEMINAR ON EIGENVALUES SEMINAR ON EIGENVALUES AND EIGENVECTORSAND EIGENVECTORS
By Vinod Srivastava M.E. Modular I & CRoll No.151522
CONTENTSCONTENTS Introduction to Eigenvalues and Eigenvectors Examples Two-dimensional matrix Three-dimensional matrix• Example using MATLAB• References
INTRODUCTIONINTRODUCTION
Eigen Vector-
In linear algebra , an eigenvector or characteristic vector of a square matrix is a vector that does not changes its direction under the associated linear transformation.
In other words – If V is a vector that is not zero, than it is an eigenvector of a square matrix A if Av is a scalar multiple of v. This condition should be written as the equation:
AV= λv
Contd….Contd….
Eigen Value-• In above equation λ is a scalar known as the eigenvalue or
characteristic value associated with eigenvector v.
• We can find the eigenvalues by determining the roots of the characteristic equation-
0 IA
ExamplesExamples
Two-dimensional matrix example-Ex.1 Find the eigenvalues and eigenvectors of matrix A.
Taking the determinant to find characteristic polynomial A-
It has roots at λ = 1 and λ = 3, which are the two eigenvalues of A.
2112
A
0IA 021
12
043 2
Eigenvectors v of this transformation satisfy the equation, Av= λvRearrange this equation to obtain-
For λ = 1, Equation becomes,
which has the solution,
0 vIA
0 vIA
00
1111
2
1
vv
1
1v
For λ = 3, Equation becomes,
which has the solution-
Thus, the vectors vλ=1 and vλ=3 are eigenvectors of A associated with the eigenvalues λ = 1 and λ = 3, respectively.
03 uIA
00
1111
2
1
uu
11
u
Three-dimensional matrix example-Ex.2 Find the eigenvalue and eigenvector of matrix A.
the matrix has the characteristics equation-
200130014
A
0234
200130
014
AI
therefore the eigen values of A are-
For λ = -2, Equation becomes,
which has the solution-
4,3,2 321
000
000110
012
0
3
2
1
11
vvv
vAI
221
v
Similarly for λ = -3 and λ = -4 the corresponding eigenvectors u and x are-
002
,011
xu
Example using MATLABExample using MATLAB
REFERENCESREFERENCES http://www.slideshare.net/shimireji Digital Control and State Variable methods by M.Gopal https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
THANKU