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Finding Eigenvalues and Eigenvectors

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  1. Finding Eigenvalues and Eigenvectors What is really important?

  2. Approaches • Find the characteristic polynomial • Leverrier’s Method • Find the largest or smallest eigenvalue • Power Method • Inverse Power Method • Find all the eigenvalues • Jacobi’s Method • Householder’s Method • QR Method • Danislevsky’s Method DRAFT Copyright, Gene A Tagliarini, PhD

  3. Finding the Characteristic Polynomial • Reduces to finding the coefficients of the polynomial for the matrix A • Recall |lI-A| = ln+anln-1+an-1ln-2+…+a2l1+a1 • Leverrier’s Method • Set Bn = A and an = -trace(Bn) • For k = (n-1) down to 1 compute • Bk = A (Bk+1 + ak+1I) • ak = - trace(Bk)/(n – k + 1) DRAFT Copyright, Gene A Tagliarini, PhD

  4. Vectors that Span a Space and Linear Combinations of Vectors • Given a set of vectors v1, v2,…, vn • The vectors are said span a space V, if given any vector xεV, there exist constants c1, c2,…, cn so that c1v1 + c2v2 +…+ cnvn = x and x is called alinear combinationof the vi DRAFT Copyright, Gene A Tagliarini, PhD

  5. Linear Independence and a Basis Given a set of vectors v1, v2,…, vn and constants c1, c2,…, cn The vectors are linearly independent if the only solution to c1v1 + c2v2 +…+ cnvn = 0 (the zero vector) is c1= c2=…=cn = 0 A linearly independent, spanning set is called a basis 5 3/12/2014 DRAFT Copyright, Gene A Tagliarini, PhD

  6. Example 1: The Standard Basis • Consider the vectors v1 = <1, 0, 0>, v2 = <0, 1, 0>, and v3 = <0, 0, 1> • Clearly, c1v1 + c2v2 + c3v3 = 0  c1= c2= c3 = 0 • Any vector <x, y, z> can be written as a linear combination of v1, v2, and v3 as<x, y, z> = x v1 + y v2 + z v3 • The collection {v1, v2, v3} is a basis for R3; indeed, it is the standard basis and is usually denoted with vector names i, j, and k, respectively. DRAFT Copyright, Gene A Tagliarini, PhD

  7. Another Definition and Some Notation • Assume that the eigenvalues for an n x n matrix A can be ordered such that|l1| > |l2| ≥ |l3| ≥ … ≥ |ln-2| ≥ |ln-1| > |ln| • Then l1 is the dominant eigenvalue and |l1| is the spectral radius of A, denoted r(A) • The ith eigenvector will be denoted using superscripts as xi, subscripts being reserved for the components of x DRAFT Copyright, Gene A Tagliarini, PhD

  8. Power Methods: The Direct Method • Assume an n x n matrix A has n linearly independent eigenvectors e1, e2,…, en ordered by decreasing eigenvalues|l1| > |l2| ≥ |l3| ≥ … ≥ |ln-2| ≥ |ln-1| > |ln| • Given any vector y0 ≠ 0, there exist constants ci, i = 1,…,n, such that y0 = c1e1 + c2e2 +…+ cnen DRAFT Copyright, Gene A Tagliarini, PhD

  9. The Direct Method (continued) • If y0 is not orthogonal to e1, i.e.,(y0)Te1≠ 0, • y1 = Ay0 = A(c1e1 + c2e2 +…+ cnen) • = Ac1e1 + Ac2e2 +…+ Acnen • = c1Ae1 + c2Ae2 +…+ cnAen • Can you simplify the previous line? DRAFT Copyright, Gene A Tagliarini, PhD

  10. The Direct Method (continued) • If y0 is not orthogonal to e1, i.e.,(y0)Te1≠ 0, • y1 = Ay0 = A(c1e1 + c2e2 +…+ cnen) • = Ac1e1 + Ac2e2 +…+ Acnen • = c1Ae1 + c2Ae2 +…+ cnAen • y1 = c1l1e1 + c2l2e2 +…+ cnlnen • What is y2 = Ay1? DRAFT Copyright, Gene A Tagliarini, PhD

  11. The Direct Method (continued) DRAFT Copyright, Gene A Tagliarini, PhD

  12. The Direct Method (continued) DRAFT Copyright, Gene A Tagliarini, PhD

  13. The Direct Method (continued) DRAFT Copyright, Gene A Tagliarini, PhD

  14. The Direct Method (continued) • Note: any nonzero multiple of an eigenvector is also an eigenvector • Why? • Suppose e is an eigenvector of A, i.e., Ae=le and c0 is a scalar such that x = ce • Ax = A(ce) = c (Ae) = c (le) = l (ce) = lx DRAFT Copyright, Gene A Tagliarini, PhD

  15. The Direct Method (continued) DRAFT Copyright, Gene A Tagliarini, PhD

  16. Direct Method (continued) • Given an eigenvector e for the matrix A • We have Ae = le and e0, so eTe  0 (a scalar) • Thus, eTAe = eTle = leTe  0 • So l = (eTAe) / (eTe) DRAFT Copyright, Gene A Tagliarini, PhD

  17. Direct Method (completed) DRAFT Copyright, Gene A Tagliarini, PhD

  18. Direct Method Algorithm DRAFT Copyright, Gene A Tagliarini, PhD

  19. Jacobi’s Method • Requires a symmetric matrix • May take numerous iterations to converge • Also requires repeated evaluation of the arctan function • Isn’t there a better way? • Yes, but we need to build some tools. DRAFT Copyright, Gene A Tagliarini, PhD

  20. What Householder’s Method Does • Preprocesses a matrix A to produce an upper-Hessenberg form B • The eigenvalues of B are related to the eigenvalues of A by a linear transformation • Typically, the eigenvalues of B are easier to obtain because the transformation simplifies computation DRAFT Copyright, Gene A Tagliarini, PhD

  21. Definition: Upper-Hessenberg Form • A matrix B is said to be in upper-Hessenberg form if it has the following structure: DRAFT Copyright, Gene A Tagliarini, PhD

  22. A Useful Matrix Construction • Assume an n x 1 vector u0 • Consider the matrix P(u) defined byP(u) = I – 2(uuT)/(uTu) • Where • I is the n x n identity matrix • (uuT) is an n x n matrix, the outer productof u with its transpose • (uTu) here denotes the trace of a 1 x 1 matrix and is the inner or dot product DRAFT Copyright, Gene A Tagliarini, PhD

  23. Properties of P(u) • P2(u) = I • The notation here P2(u) = P(u) * P(u) • Can you show that P2(u) = I? • P-1(u) = P(u) • P(u) is its own inverse • PT(u) = P(u) • P(u) is its own transpose • Why? • P(u) is an orthogonal matrix DRAFT Copyright, Gene A Tagliarini, PhD

  24. Householder’s Algorithm • Set Q = I, where I is an n x n identity matrix • For k = 1 to n-2 • a = sgn(Ak+1,k)sqrt((Ak+1,k)2+ (Ak+2,k)2+…+ (An,k)2) • uT = [0, 0, …, Ak+1,k+ a, Ak+2,k,…, An,k] • P = I – 2(uuT)/(uTu) • Q = QP • A = PAP • Set B = A DRAFT Copyright, Gene A Tagliarini, PhD

  25. Example DRAFT Copyright, Gene A Tagliarini, PhD

  26. Example DRAFT Copyright, Gene A Tagliarini, PhD

  27. Example DRAFT Copyright, Gene A Tagliarini, PhD

  28. Example DRAFT Copyright, Gene A Tagliarini, PhD

  29. Example DRAFT Copyright, Gene A Tagliarini, PhD

  30. Example DRAFT Copyright, Gene A Tagliarini, PhD

  31. Example DRAFT Copyright, Gene A Tagliarini, PhD

  32. How Does It Work? • Householder’s algorithm uses a sequence of similarity transformationsB = P(uk) A P(uk)to create zeros below the first sub-diagonal • uk=[0, 0, …, Ak+1,k+ a, Ak+2,k,…, An,k]T • a = sgn(Ak+1,k)sqrt((Ak+1,k)2+ (Ak+2,k)2+…+ (An,k)2) • By definition, • sgn(x) = 1, if x≥0 and • sgn(x) = -1, if x<0 DRAFT Copyright, Gene A Tagliarini, PhD

  33. How Does It Work? (continued) • The matrix Q is orthogonal • the matrices P are orthogonal • Q is a product of the matrices P • The product of orthogonal matrices is an orthogonal matrix • B = QTA Q hence Q B = Q QTA Q = A Q • Q QT = I (by the orthogonality of Q) DRAFT Copyright, Gene A Tagliarini, PhD

  34. How Does It Work? (continued) • If ek is an eigenvector of B with eigenvalue lk, then B ek =lkek • Since Q B = A Q,A (Qek) = Q (B ek) = Q (lk ek) = lk(Q ek) • Note from this: • lkis an eigenvalue of A • Qek is the corresponding eigenvector of A DRAFT Copyright, Gene A Tagliarini, PhD

  35. The QR Method: Start-up • Given a matrix A • Apply Householder’s Algorithm to obtain a matrix B in upper-Hessenberg form • Select e>0 and m>0 • e is a acceptable proximity to zero for sub-diagonal elements • m is an iteration limit DRAFT Copyright, Gene A Tagliarini, PhD

  36. The QR Method: Main Loop DRAFT Copyright, Gene A Tagliarini, PhD

  37. The QR Method: Finding The l’s DRAFT Copyright, Gene A Tagliarini, PhD

  38. Details Of The Eigenvalue Formulae DRAFT Copyright, Gene A Tagliarini, PhD

  39. Details Of The Eigenvalue Formulae DRAFT Copyright, Gene A Tagliarini, PhD

  40. Finding Roots of Polynomials • Every n x n matrix has a characteristic polynomial • Every polynomial has a corresponding n x n matrix for which it is the characteristic polynomial • Thus, polynomial root finding is equivalent to finding eigenvalues DRAFT Copyright, Gene A Tagliarini, PhD

  41. Example Please!?!?!? • Consider the monic polynomial of degree nf(x) = a1 +a2x+a3x2+…+anxn-1 +xn and the companion matrix DRAFT Copyright, Gene A Tagliarini, PhD

  42. Find The Eigenvalues of the Companion Matrix DRAFT Copyright, Gene A Tagliarini, PhD

  43. Find The Eigenvalues of the Companion Matrix DRAFT Copyright, Gene A Tagliarini, PhD