Eigenvalues and Eigenvectors
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The document is a seminar presentation by Vinod Srivastava covering the concepts of eigenvalues and eigenvectors in linear algebra, explaining their definitions, relationships, and how to calculate them with examples of two-dimensional and three-dimensional matrices. It provides step-by-step calculations using matrices to derive both eigenvalues and their corresponding eigenvectors. The presentation also includes a MATLAB example and references for further reading.
Eigenvalues and Eigenvectors
- 1. SEMINAR ON EIGENVALUESSEMINAR ON EIGENVALUES AND EIGENVECTORSAND EIGENVECTORS By Vinod Srivastava M.E. Modular I & C Roll No.151522
- 2. CONTENTSCONTENTS Introduction to Eigenvalues and Eigenvectors Examples Two-dimensional matrix Three-dimensional matrix • Example using MATLAB • References
- 3. 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
- 4. 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 λ
- 5. 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. = 21 12 A ⇒=− 0IA λ 0 21 12 = − − λ λ 043 2 =+−⇒ λλ
- 6. Eigenvectors v of this transformation satisfy the equation, Av= λv Rearrange this equation to obtain- For λ = 1, Equation becomes, which has the solution, ( ) 0=− vIA λ ( ) 0=− vIA = 0 0 11 11 2 1 v v − = 1 1 v
- 7. 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 = − − 0 0 11 11 2 1 u u = 1 1 u
- 8. Three-dimensional matrix example- Ex.2 Find the eigenvalue and eigenvector of matrix A. the matrix has the characteristics equation- − − − = 200 130 014 A ( )( )( ) 0234 200 130 014 =+++= + −+ −+ =− λλλ λ λ λ λ AI
- 9. therefore the eigen values of A are- For λ = -2, Equation becomes, which has the solution- 4,3,2 321 −=−=−= λλλ ( ) = − − =− 0 0 0 000 110 012 0 3 2 1 11 v v v vAIλ = 2 2 1 v
- 10. Similarly for λ = -3 and λ = -4 the corresponding eigenvectors u and x are- = = 0 0 2 , 0 1 1 xu
- 11. Example using MATLABExample using MATLAB
- 12. REFERENCESREFERENCES http://www.slideshare.net/shimireji Digital Control and State Variable methods by M.Gopal https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
- 13. THANKU