Eigen value and vector of linear transformation.pptx

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1. Institute for Excellence in Higher Education,Bhopal Sessions:- 2022-2023 Topic:- Eigen values and Eigen vectors of linear transformation Submitted To: Dr.Manoj Ughade Sir (Professor in Department of Mathematics,IEHE ) Submitted By:- Atul (420509) and Dilpreet Sandhu (420513) B.Sc (H) Physics III year

2. ACKNOWLEDGEMENT We ( Atul and Dilpreet Sandhu) would like to express my special thanks of gratitude to our teacher (PROF. Manoj Ughade Sir) who gave me the golden opportunity to do this wonderful assignment on the topic (Eigen values and Eigen vectors of linear transformation ), which also helped me in doing a lot of Research and I came to know about so many new things. I am really thankful to him.

Secondly, I would also like to thank my parents and friends who helped me a lot in finalizing this project within the limited time frame.

3. CERTIFICATE This is to certify that Atul and Dilpreet Sandhu, a student of B.Sc.(Physics Honours,VI sem ) acquaring roll number – 420509 and 420513 respectively, has successfully completed the assignment of “Mathematics” on the topic “Eigen values and Eigen vectors of linear transformation” under the guidance of Manoj Ughade Sir( PROF.of Mathematics department in INSTITUTE FOR EXCELLENCE IN HIGHER EDUCATION ,BHOPAL). TEACHER SIGN. SUBMITTED BY Atul (420509) and Dilpreet Sandhu(420513) Date:. 31/03/2023

4. EIGENVALUES & EIGENVECTORS OF LINEAR TRANSFORMATIONS. Eigenvalues and eigenvectors are important concepts in linear algebra that arise when studying linear transformations. EIGEN VALUES:- An eigenvalue of a linear transformation is a scalar that represents how the transformation scales a given eigenvector. In other words, when the linear transformation is applied to an eigenvector, the resulting vector is a scalar multiple of the original vector, with the scalar being the eigenvalue. EIGEN VECTORS:- An eigenvector of a linear transformation is a non-zero vector that, when multiplied by the transformation, yields a scalar multiple of itself. In other words, the transformation only changes the scale of the eigenvector, not its direction.

5. If A is an n  n matrix, do there exist nonzero vectors x in R n such that A x is a scalar multiple of x ？ Geometrical Interpretation Eigenvalue and eigenvector: A ： an n  n matrix  ： a scalar x ： a nonzero vector in R n Eigenvalue Eigenvector

6. Ex 1: (Verifying eigenvalues and eigenvectors) Eigenvalue Eigenvector Eigenvalue Eigenvector

7. Thm 1 : (The eigenspace of A corresponding to  ) If A is an n  n matrix with an eigenvalue , then the set of all eigenvectors of  together with the zero vector is a subspace of R n . This subspace is called the eigenspace of  . Pf: x 1 and x 2 are eigenvectors corresponding to 

8. Mathematical defination for Eigenvalues and eigenvectors of linear transformations:

9. Thm 2 : (Finding eigenvalues and eigenvectors of a matrix A  M n  n ) Let A is an n  n matrix. (2) The eigenvectors of A corresponding to  are the nonzero solutions of (1) An eigenvalue of A is a scalar  such that If has nonzero solutions iff . Note: (homogeneous system) Characteristic polynomial of A  M n  n : Characteristic equation of A :

10. Ex 2 :(Finding eigenvalues and eigenvectors) Sol: Characteristic equation:

11. Ex 3 : (Finding eigenvalues and eigenvectors) Find the eigenvalues and corresponding eigenvectors for the matrix A . What is the dimension of the eigenspace of each eigenvalue? Sol: Characteristic equation: Eigenvalue:

12. The eigenspace of A corresponding to : Thus, the dimension of its eigenspace is 2.

13. Notes: (1) If an eigenvalue  1 occurs as a multiple root ( k times ) for the characteristic polynominal, then  1 has multiplicity k. (2) The multiplicity of an eigenvalue is greater than or equal to the dimension of its eigenspace.

14. Ex 4 ： Find the eigenvalues of the matrix A and find a basis for each of the corresponding eigenspaces . Sol: Characteristic equation:

15. is a basis for the eigenspace of A corresponding to

16. is a basis for the eigenspace of A corresponding to

17. is a basis for the eigenspace of A corresponding to

18. Thm 3: (Eigenvalues of triangular matrices) If A is an n  n triangular matrix, then its eigenvalues are the entries on its main diagonal. Ex 5 : (Finding eigenvalues for diagonal and triangular matrices) Sol:

19. Ex 6: (Finding eigenvalues and eigenspaces ) Sol:

20. Eigenvalues and eigenvectors have many applications in various fields, including mathematics, physics, engineering, computer science, and more. Here are some specific examples: Image and signal processing: Eigenvectors and eigenvalues are used in image and signal processing techniques, such as Principal Component Analysis (PCA), which is used to reduce the dimensionality of data sets while retaining as much variance as possible. In this application, the eigenvectors of the covariance matrix of a set of data points are used as a basis for a new coordinate system that captures the most important features of the data. Quantum mechanics: Eigenvectors and eigenvalues are used to represent the state of a quantum system. In quantum mechanics, the wave function of a particle is an eigenvector of the Hamiltonian operator, and the corresponding eigenvalue represents the energy of the system. Structural engineering: Eigenvectors and eigenvalues are used in the analysis of structures, such as bridges and buildings, to determine their natural frequencies and modes of vibration. This information can be used to design structures that can withstand external forces, such as wind and earthquakes. These are just a few examples of the many applications of eigenvectors and eigenvalues. They are powerful tools that can be used to solve a wide range of problems in various fields. Applications

21. REFERENCES ocw.mit.edu “Matrices & Tensors in Physics” By AW Joshi REFERENCES 2

22. THANK YOU Presentation by Dilpreet Sandhu(420513) And Atul(4205 09 ).