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THE EIGENVALUE PROBLEM

THE EIGENVALUE PROBLEM. BY YAN RU LIN SCOTT HENDERSON NIRUPAMA GOPALASWAMI GROUP 4. 11.1 EIGENVALUES & EIGENVECTORS. Definition. An eigenvector of a n x n matrix A is a nonzero vector x such that for some scalar .

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THE EIGENVALUE PROBLEM

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  1. THE EIGENVALUE PROBLEM BY YAN RU LIN SCOTT HENDERSON NIRUPAMA GOPALASWAMI GROUP 4

  2. 11.1 EIGENVALUES & EIGENVECTORS

  3. Definition • An eigenvector of a n x n matrix A is a nonzero vector x such that for some scalar . • A scalar is called an eigenvalue of A if there is a nontrivial solution x of ; such an x is called an eigenvector corresponding to . D.C. Lay, "Eigenvectors and Eigenvalues," in Linear Algebra and Its Applications, 3rd ed. Boston, MA: Pearson, 2006, ch. 5, pp. 301-372

  4. Formulation • This is also equivalent to • Now it can be solved easily using a determinant Essentially, for some value of λ given a transformation matrix A, there may exist a vector x such that the equation is satisfied. If λ and this particular vector x do exist, then we call λ the eigenvalue and the x the corresponding eigenvector.

  5. Example • Solve now using a determinant

  6. Importance • Eigenvalues and eigenvectors find numerous applications in these areas: • Differential Equations • Dynamical systems • Engineering design • Chemistry and physics • Schrödinger equation (quantum mechanics) • Vibration analysis

  7. BRAINBITE A. λ=2 B. λ=0 C. λ=-3 D. λ=1 Remember that… x is given along with A, so λ is solved easily http://www.maths.usyd.edu.au/u/UG/JM/MATH1014/Quizzes/quiz10.html Answer: C

  8. 11.2 EIGENVALUES SOLUTION PROCEDURE AND APPLICATIONS

  9. 11.2 Eigenvalues Solution Procedure and Applications • Ax = lx  (A-lI)x = 0 • x=0 is a trivial solution • Non-trivial solutions exist if and only if:

  10. Resulting algebraic equation is called the characteristic equation. • Characteristic polynomial- nth-order polynomial in l • Roots are the eigenvalues{l1, l2, …, ln} • Solution space is called eigenspace corresponding to {l1, l2, …, ln} • The solutions obtained are called eigenvectors

  11. Eigenvalue Example • Characteristic matrix • Characteristic equation • Eigenvalues: l1 = -5, l2 = 2

  12. 11.2 Subsection(1) -Quick Tips • An n x n matrix A means that are n values to x, and there will be n eigenvectors and eigenvalues even if some are duplicated • The eigenvalues of a triangular matrix are the entries on its main diagonal • Consider that since λis scalar, A must act on eigenvectors only to “stretch” x and not to change its direction (see figure) Unknown. (2011, Oct 27). Eigenvalues and eigenvectors [Online]. Available: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

  13. Example • Click here to view a demo on eigenvalues and eigenvectors • http://web.mit.edu/18.06/www/Demos/eigen-applet-all/eigen_sound_all.html

  14. 11.2 Subsection(2)-Determining Eigenvectors • First determine eigenvalues: {l1, l2, …, ln} • Then determine eigenvector corresponding to each eigenvalue: • Eigenvectors determined up to scalar multiple • Distinct eigenvalues • Produce linearly independent eigenvectors • Repeated eigenvalues • Produce linearly dependent eigenvectors • If n roots are equal then the eigenvalues are said to of multiplicity n.

  15. Eigenvector Example • Eigenvalues • Determine eigenvectors: Ax = lx • Eigenvector for l1 = -5 • Eigenvector for l1 = 2

  16. BRAINBITE http://www.maths.usyd.edu.au/u/UG/JM/MATH1014/Quizzes/quiz12.html Answer : c

  17. 11.2.2 APPLICATIONS TO ELEMENTARY SINGULARITIES IN THE PHASE PLANE • Consider a linear system of ODEs given by • If the eigenvalues λ is real

  18. If eigenvalues are complex of the form

  19. 11.2. subsection(3)Special matrices in exercises(1) Markov Matrix • Let A= • The sum of elements of row or column sum to unity. • One of the eigenvalue of Markov matrix is 1. • The rows of [A-I]sum to zero • [A-I] is singular and columns of A-I are linearly dependent. M.D. Greenberg, "The Eigenvalue Problem," in Advanced Engineering Mathematics, 2nd ed. Upper Saddle River, New Jersey: Prentice Hall, 1998, ch.11.3

  20. (2)Tridiagnol matrix • A Tridiagnol matrix is one in which all element are zero except the principal diagonals and its two adjacent diagonals . • Eigenvalues are given by

  21. (3) Generalized eigenvalue problem • If B≠1 then Ax=λBx is called generalized eigenvalue problem. • Characteristic equation got by det(A - λB)x=0 • Eigenvectors given by (A - λB)x=0

  22. (4) Cayley hamilton theorem • Theorem- The characteristic equation of any square matrix A is λn+ α1 λn-1 +…. αn λ =0 then An+ α1 An-1+…+ αn -1A+ αn I=0. i.eA satisfies characteristic equation.

  23. BRAINBITE http://www.maths.usyd.edu.au/u/UG/JM/MATH1014/Quizzes/quiz12.html Answer : a

  24. 11.3 SYMMETRIC MATRICES

  25. A square matrix is symmetric if A = AT. This means that each element aij = aji, as figure[1]. • A symmetric matrix needs not have real numbers as elements. However, when it does, it has the remarkable property of having only real eigenvalues. • Proof : recall, a complex number, then, the conjugate of z is defined to be , and , …[2] [1] http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_symmetric_matrix.htm [2]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.3, pp. 354-362.

  26. Let A be an n x n matrix. Let be the eigenvector corresponding to its eigenvalue , and get , , If A is a real and symmetric matrix, then and , now compute[1]… [1]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.3, pp. 354-362.

  27. is a 1x1 matrix (a number), and so is the same as its transpose Therefore, the number , being equal to its conjugate, is a real number. And is certainly real. Thereforeis real[1]. [1]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.3, pp. 354-362.

  28. Let A be a real symmetric matrix. Then eigenvectors associated with distinct eigenvalues are orthogonal. • Let A be a real symmetric matrix. Then there is a real, orthogonal matrix that diagonalizesA[1]. • Let A be a real, n x n symmetric matrix. Then its eigenvector provide an orthogonal basis for n-space. Therefore, if an eigenvalue is repeated by k times. Then the eigenspace is of dimension k, and we can find another set of orthogonal vector by linear combination[2]. [1] Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.3, pp. 354-362. [2] M.D. Greenberg, "The Eigenvalue Problem," in Advanced Engineering Mathematics, 2nd ed. Upper Saddle River, New Jersey: Prentice Hall, 1998, ch.11.3, pp. 554-569.

  29. Symmetric Matrix Examples • We can see that a real, symmetric matrix provides a set of real eigenvalues. And the corresponding eigenvectors are • These form an orthogonal set of vectors[1]. [1]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.3, pp. 354-362.

  30. The orthonormal form is divided by its length and they can be used as columns of an orthogonal matrix. • We can find Q-1 = QT, and A can be diagonalized by Q[1]. [1]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.3, pp. 354-362.

  31. Useful properties[1] • If A, B are symmetric n × n matrices, then A+B is symmetric. • If A, B are symmetric n × n matrices, then AB is not symmetric. • If C is any n × n matrix. Then is a symmetric matrix. = • If D is a diagonal matrix, then D is symmetric. [1]http://www.math.panam.edu/

  32. BRAINBITE Find which of followings is not the eigenvalue of the 4x4 matrix and its corresponding orthogonal eigenvector. a. b. c. d. e. [1]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.3, pp. 354-362. answer : d

  33. 11.4 DIAGONALIZATION

  34. Background • Diagonal matrices have good properties for simplifying calculations • Exploit these properties by diagonalizingthe matrix A in or for DE • Essentially, a change of base is required () so that…

  35. Properties and Restrictions • A is diagonalizable if and only if it has n LI eigenvectors • If the above condition is met, (i.e. the eigenvectors of A form a matrix Q) • Symmetric matrices are always diagonalizable. • due to property of LI eigenvectors

  36. Example • where and • Make substitution of • where • Solve for eigenvalues and eigenvectors to find Dand Q

  37. Example • Eigenvalues are of A are and • Eigenvectors are and • Therefore, when

  38. Example • Using D, two uncoupled differential equations arise (instead of coupled like before) • These equations are simple ODE and are solved using the solution . • ω can be solved for easily whereas A and φare constants of integration Diagonalization

  39. Example • Since we assumed , now we can solve for the real x(t) and y(t) • x(t) and y(t) have been solved completely and easily compared to not using properties of diagonal matrices

  40. 11.5 Applications to first order systems with constant coefficients

  41. 11.5 Applications to first order systems with constant coefficients • Consider an initial value problem • In matrix form

  42. The solution to the differential equation is given by • Where A= coefficients of variables Q= modal matrix=[e1,e2….en] D= Diagonal matrix where jth diagonal elements are jth eigenvalue of A.

  43. The solution can also be expressed of the form Where M.D. Greenberg, "The Eigenvalue Problem," in Advanced Engineering Mathematics, 2nd ed. Upper Saddle River, New Jersey: Prentice Hall, 1998, ch.11.3

  44. Example • Consider the equations • Solution :

  45. Replacing the values of A,D,Q and Q-1 in the following equation • we get

  46. 11.6 QUADRATIC FORMS

  47. A (complex) quadratic form is an expansion , in which each and is a complex number[1]. For n=2, this is • The quadratic form is real if each and is real, and we usually write as , and the form is . [1]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.4, pp. 363-367.

  48. It is often convenient to write a quadratic form in a matrix form. If A = [], then =[1] [1]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.4, pp. 363-367.

  49. Quadratic forms example • Let then But we can also rewrite the quadratic form as = The advantage of latter form is that A is a symmetric matrix[1]. [1]Peter V. O’Neil, “Eigenvalues,Diagnolization and Special Matrices” in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch.8.4, pp. 363-367.

  50. Classification of The Quadratic Form[1] : A quadratic form is said to be: • Negative definite: when • Negative semidefinite:for all x and for some • Positive definite:when • Positive semidefinite:for all x and for some • Indefinite:for some x and for some other x [1]http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CGQQFjAE&url=http%3A%2F%2Fwww.econ.iastate.edu%2Fclasses%2Fecon501%2FHallam%2Fdocuments%2FQuad_Forms_000.pdf&ei=wp26TvrgMsiJsAKNzqXOCA&usg=AFQjCNGQ_OibQn6rhf0wrBTNSVMVOltoaQ&sig2=Y061Hf2_fqbXqjUYyADczQ

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