Linear Algebra : Matrix Eigenvalue Problems – Part 1 By Dr. Samer Awad

1. Linear Algebra: Matrix Eigenvalue Problems – Part 1 By Dr. SamerAwad Assistant professor of biomedical engineering The Hashemite University, Zarqa, Jordan samer.awad@gmail.com Last update: 28 March 2016

2. 2 samer.awad@hu.edu.jo 28 March 2016 Eigenvalue Problems • • The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering. • • Solving eigenvalue problems is equivalent to matrix diagonalization and has several applications: • Stability analysis • The physics of rotating bodies • Small oscillations of vibrating systems • • You might encounter these and/or other applications of eigenvalue problems in other courses.

3. 3 samer.awad@hu.edu.jo 28 March 2016 Matrix Eigenvalue Problems • A matrix eigenvalue problem considers the vector equation: Here, A is a givensquare matrix, is an unknown scalar is an unknown vector • In a matrix eigenvalue problem, the task is to determine: ‒ ’s that satisfy the eq. above (called eigenvalues). ‒’s that satisfy the eq. above (called eigenvectors) excluding which is always a solution.

4. 4 samer.awad@hu.edu.jo 28 March 2016 Matrix Eigenvalue Problems • The set of all the eigenvalues of A is called the spectrum of A. • The spectrum consists of at least one eigenvalue and at most of n numerically different eigenvalues.

5. 5 samer.awad@hu.edu.jo 28 March 2016 Eigenvalue Problems Steps • Steps for solving Eigenvalue Problems: ‒ Solve the c/s equation(solutions are eigenvalues): ‒Substitute each into ‒Solve the system of linear equations (i.e. find for each). These vectors are the eigenvectors). ‒ You can always check your solution by substituting your and the corresponding into:

6. 6 samer.awad@hu.edu.jo 28 March 2016 Example 1: Finding Eigenvalues and Eigenvectors • For the matrix above, eigenvaluesmust be determined first: • Transferring the terms on the right to the left, we get

7. 7 samer.awad@hu.edu.jo 28 March 2016 Example 1: Finding Eigenvalues and Eigenvectors • This can be written in matrix notation: Proof:  • So we’ve transferred the original eigenvalue equation to a homogeneous linear system. • By Cramer’s rule, has a nontrivial solution iff its coefficient determinant (det) is zero.

8. 8 samer.awad@hu.edu.jo 28 March 2016 Example 1: Finding Eigenvalues and Eigenvectors • is the characteristic determinant (or characteristic polynomial) and is the characteristic equation of A. • Solving the characteristic equations gives the two eigenvalues: • Solution of quadratic equation:

9. 9 samer.awad@hu.edu.jo 28 March 2016 Example 1: Finding Eigenvalues and Eigenvectors • Eigenvector xof A corresponding to 𝜆 can be obtained from: • by substituting: • Gauss elimination will zero row 2 which means we have infinite solutions. Rearranging row 1 or row 2 gives the solution:

10. 10 samer.awad@hu.edu.jo 28 March 2016 Example 1: Finding Eigenvalues and Eigenvectors • solution: • Hence for 𝜆 • If we choose we obtain the eigenvector • Check:

11. 11 samer.awad@hu.edu.jo 28 March 2016 Example 1: Finding Eigenvalues and Eigenvectors • Eigenvector xof A corresponding to can be obtained from: • by substituting : • Gauss elimination will also zero row 2 which means we have infinite solutions. Rearranging row 1 or row 2 gives the solution:

12. 12 samer.awad@hu.edu.jo 28 March 2016 Example 1: Finding Eigenvalues and Eigenvectors • solution: • Hence for 𝜆 • If we choose we obtain the eigenvector • Check:

13. 13 samer.awad@hu.edu.jo 28 March 2016 Finding Eigenvalues and Eigenvectors: General Case • Transferring the terms on the right side to the left side: • Which is equivalent to:

14. 14 samer.awad@hu.edu.jo 28 March 2016 Finding Eigenvalues and Eigenvectors: General Case • By Cramer’s theorem this homogeneous linear system of equations has a nontrivial solution if and only if the corresponding determinant of the coefficients is zero: • Which is equivalent to:

15. 15 samer.awad@hu.edu.jo 28 March 2016 Eigenvalue Problems Steps • Steps for solving Eigenvalue Problems: ‒ Solve the c/s equation(solutions are eigenvalues): ‒Substitute each into ‒Solve the system of linear equations (i.e. find for each). These vectors are the eigenvectors). ‒ You can always check your solution by substituting your and the corresponding into:

16. 16 samer.awad@hu.edu.jo 28 March 2016 Definitions • is called the characteristic matrix. • is called thecharacteristic determinant of A. •is called the characteristic equation of A. • By developing we obtain a polynomial of nth degree in . This is called the characteristic polynomial of A. • Theorem 1: Eigenvalues: The eigenvalues of a square matrix A are the roots of the characteristic equation A. • Hence an n x n matrix has at least one eigenvalue and at most n different eigenvalues.

17. 17 samer.awad@hu.edu.jo 28 March 2016 Definitions • Theorem 2: Eigenvectors, Eigenspace: If w and x are eigenvectors of a matrix A corresponding to the same eigenvalue , so are w + x (provided x ≠ ‒w) and kx for any k≠0. • Hence the eigenvectors corresponding to one and the same eigenvalue of A, together with 0, form a vector space, called the eigenspace of A corresponding to that .

18. 18 samer.awad@hu.edu.jo 28 March 2016 Example 2: Multiple Eigenvalues • Find the eigenvalues and eigenvectors of • For our matrix, the characteristic determinant gives the characteristic equation: • The roots (eigenvaluesof A) are:

19. 19 samer.awad@hu.edu.jo 28 March 2016 Example 2: Multiple Eigenvalues • For the characteristic matrix is:  • After two steps of gauss elimination:

20. 20 samer.awad@hu.edu.jo 28 March 2016 Example 2: Multiple Eigenvalues • For : • From row 2  • From row 1 &  • Hence for 𝜆 • If we choose we obtain

21. 21 samer.awad@hu.edu.jo 28 March 2016 Example 2: Multiple Eigenvalues • For the characteristic matrix is:  • After two steps of gauss elimination:

22. 22 samer.awad@hu.edu.jo 28 March 2016 Example 2: Multiple Eigenvalues • For : • From row 1 , • Hence for 𝜆 • If we choose we obtain

23. 23 samer.awad@hu.edu.jo 28 March 2016 Example 5: Real Matrices with Complex Eigenvalues & Eigenvectors • Find the eigenvalues and eigenvectors of the following skew-symmetric matrix: • Solving the characteristic equations gives the two eigenvalues:.

24. 24 samer.awad@hu.edu.jo 28 March 2016 • Example 5: Real Matrices with Complex Eigenvalues & Eigenvectors • Eigenvector of A corresponding to 𝜆 can be obtained from: Gauss elimination  • From row 1  • Hence for 𝜆 • If we choose we obtain the eigenvector

25. 25 samer.awad@hu.edu.jo 28 March 2016 • Example 5: Real Matrices with Complex Eigenvalues & Eigenvectors • Eigenvector of A corresponding to 𝜆 can be obtained from: Gauss elimination  • From row 1  • Hence for 𝜆 • If we choose we obtain the eigenvector

26. 26 samer.awad@hu.edu.jo 28 March 2016 Eigenvalues of The Transpose of a Matrix • Theorem3: Eigenvalues of the Transpose: The transpose AT of a square matrix A has the same eigenvalues as A.