1. OCE301 Part II: Linear Algebra lecture 4

2. Eigenvalue Problem Ax = y Ax = x occur frequently in engineering analysis (eigenvalue problem) Ax =  x [ A -  x = 0 K = [ A -  characteristic matrix of the matrix A a set of linear simultaneous algebraic equations homogeneous equations y = x if K  x = 0

3. Homogeneous Equations Kx = 0 trivial solution x = 0 (if det K is not equal to zero, from Cramer’s rule) nontrivial solution can occur if det K = 0 (i.e. if K is singular)

4. Cramer’s Rule: General D k is the determinant obtained from D by replacing in D the kth column by the column with the entries b 1, …, b n.

5. Characteristic Equation K = [ A -  [ A -  x = 0 nontrivial solution can occur if det K = 0 (i.e. if K is singular) D( ) = det K D( ) = det (A -  characteristic determinant characteristic equation of the matrix A

6. D( ): Polynomial in D( ) = det (A -  D( ) =  n n +  n-1 n-1 +  n-2 n-2 + … +  1 +  0 The characteristic determinant D( ) = det (A -  is clearly a polynomial in :

7. Characteristic Values (Eigenvalues) D( ) =  n n +  n-1 n-1 +  n-2 n-2 + … +  1 +  0 = 0 characteristic equation the roots are called characteristic values or eignevalues The set of the eigenvalues is called the spectrum of A Ax = x The largest of the absolute values of the eigenvalues of A is called the spectral radius of A

8. Eigen Vectors Kx = 0 Once eigenvalues ( ) are known, corresponding eigenvectors (x) are obtained from the above system.

9. Features of Eigenvectors Ax = x A (kx) = (kx) implies If x is an eigenvector of a matrix A corresponding to an eigenvalue , so is kx with any k not equal to zero. All eigenvectors that derived from unequal eigenvalues are linearly independent.

10. Example: Eigen values  1 = -1, 2 = -2, 3 = -3

11. Example: Eigen vectors  using 1 = -1 [ A – 1  x 1 = 0

12. Example: Eigen vectors (contd.) x 1 = - x 2 and x 3 = - x 2  2 = -2  3 = -3  1 = -1 (may apply row operations to obtain a row-equivalent system)

13. Matlab function: eig eig Eigenvalues and Eigenvectors. E = eig(X) is a vector containing the eigenvalues of a square matrix X. [V,D] = eig(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D.

14. Matlab Example » a=[ 0 1 0; 0 0 1; -6 -11 -6] a = 0 1 0 0 0 1 -6 -11 -6 » e=eig(a) e = -2.0000 -3.0000 » [v,d]=eig(a) v = -0.5774 0.2182 -0.1048 0.5774 -0.4364 0.3145 -0.5774 0.8729 -0.9435 d = -1.0000 0 0 0 -2.0000 0 0 0 -3.0000 » diag(d)' ans = -1.0000 -2.0000 -3.0000

15. Matlab Example (contd.) » v v = -0.5774 0.2182 -0.1048 0.5774 -0.4364 0.3145 -0.5774 0.8729 -0.9435 » v(:,1)/v(2,1) ans = 1.0000 » v(:,2)/v(2,2) ans = -0.5000 1.0000 -2.0000 » v(:,3)/v(2,3) ans = -0.3333 1.0000 -3.0000

16. Definitions: Symmetric etc. symmetric A T = A skew-symmetric A T = - A orthogonal A T = A -1 A = a real square matrix

17. Eigenvalues of Symmetric, Skew- symmetric and Orthogonal Matrices The eigenvalues of a symmetric matrix are real. The eigenvalues of a skew-symmetric matrix are pure imaginary or zero. The eigenvalues of an orthogonal matrix are real or complex conjugates in pairs and have the absolute value 1 ( | | = 1 ).

18. Inner (dot) Product of Vectors inner product or dot product

19. Orthogonal Matrix = I it means A T = A -1

20. Orthonormal System A real square matrix is orthogonal if and only if its column vectors a 1, …, a n form an orthonormal system, that is

21. Definitions: Hermitian etc. symmetric A T = A skew-symmetric A T = - A orthogonal A T = A -1 Hermitian skew-Hermitian unitary A T = A A T = - A A T = A -1 Real square matrix Complex square matrix : replacing each entry by its complex conjugate A generalized

22. Eigenvalues of Hermitian, Skew- Hermitian and Unitary Matrices The eigenvalues of a Hermitian matrix are real. The eigenvalues of a skew-Hermitian matrix are pure imaginary or zero. The eigenvalues of a unitary matrix have the absolute value 1 ( | | = 1 ).

23. Location of the Eigenvalues Hermitian Unitary Skew-Hermitian 1 Re Im

24. Eigenvalues of Hermitian Proof: the eigenvalues of a Hermitian matrix are real. A T = A (a real number) real a scalar alternative form