5.3 - 324 Chapter 5 Linear Transformations We have learned...

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Unformatted text preview: 324 Chapter 5 Linear Transformations We have learned to compute eigenvalues, eigenvectors, and eigenspaces for a square matrix, and have related these to the linear transformation defined by the matrix: the Iransfonnation reduces to multiplic‘atz’on by a scalar - ' on each eigenvector The characteristic roOts of a linear second— order DE with constant coefficients turn out __to Q be the eigenvalues of the matrix of the system to which it corresponds 5.3 Problems Computing Eigenstuff For each matrix in Problems Im16, compare its eigenvalues and eigenvectorfs), and sketch the eigenspac-es when the eigerwectors are real. 17. 18. l3 ‘31 2- l3 El [—2151 5H .m] r: 1] .M am] r: a {-2 n r. :l l—? —‘2‘l l3 3i [i ”ll Eigenvector Shortcut For a 2 x 2 matrix a b A = ls l with eigenvalue A, show that if [3 ¢ 0, then the corre— sponding eigenveetor is s a [ail]. When Shortcut Fails The eigenvector shortcut of Prob~ lem 17 may fail when I; = 0, forcing a return to the deft» nition Air 2 A?! to find the eigenvectods). For each eigenv value in the following matrices, find the eigenvectods). Discuss how/why the shortcut fails and why the definition succeeds. [z 2] [2 2] [3 .2] More Eigenstuff For each matrix in Problems 19-34, com~ pure its eigenvalues, eigenvectors and the dimension of each eigenspac-e. 19. .31. 33. 35. .36. 2 O O l v—1 —2 29. ml 0 l I 2 2 0 l —-l 2 0 3 22. 0 —l 1 2 3 0 G 0 0 0 l l l 0 0 l 0 1 24. ——1 3 0 1 i 0 3 2 —2 ml 0 l 2 2 3 mi 3 0 26. 1 2 1 .24 13 ~1 2 “2 1 1 O 0 l l 1 —4 3 0 28. 0 l 1 ~4 2 1 0 0 i 3 —2 O 0 D 2 I O O 30. —1 l 2 —t I 3 ——i 0 3 2 I 8 ~1 4 D 4 0 0 4 0 O 32 O 4 0 0 0 0 6 O " 0 0 8 0 0 0 O 4 —l —2 l 8 2 O i 2 2 O O 0 0 2 0 O 34 l w2 0 0 0 0 6 0 ' 1 0 1 O 0 O l 4 0 2 0 l Prove the Eigenspace Theorem Show that the set of eigenvectors belonging to a particular eigenvalue of an n x :1 matrix, together with the zero vector, is a subspace of til". HINT: Use equation (2) and verify closure. Distinct Eigenvaiues Extended Extend the proof of the Distinct Eigenvaiue Theorem for a 3 x 3 matrix A as fol- lows: Show that if A has 3 distinct eigenvalues Ag, A3, Ag, then the corresponding eigenvectors in, i5, v3 are linearly independent. HINT: Use the fact that an eigenveetor v, can— not be zero, and follow the steps shown in the proof for two distinct eigenvalues. Section 5.3 Eigenvalues and Eigenvectors 325 Invertible Matrices 37. Show that an invertible matrix cannot have a zero eigen— value In fact, you will have proved a characteristic of invertibie matrices. 38. Suppose that l is an eigenvalue of an invertible matrix A. Show that l/lt is an eigenvalue of A”. 39. Give an example to illustrate Problem 38. 40. Similar Matrices (a) Use the definition for similar matrices tie, B w A it and only if B m ?“‘AP for some invertible ma- trix P) to show that similar matrices have the same characteristic polynomials and eigenvalues (b) Show, using 2 x 2 matrices as examples, that the eigen— vectors may be different for similar matrices. 41. Identity Eigenstuff What are the eigenvalues and eigen~ vectors of the following? 10 Ig=[01]. What about I3? What about I“? 42. Eigenvaiues and Inversion if a matrix A has an inverse A‘l , use equation (2) to show that A”1 has the same eigen- vectors as A. Determine a r‘eiationship between the eigen— values oi' A and 23*. Illustrate with a suitable example Triangular Matrices The eigenvalues of an upper triangtt» lar matrix and those ofa lower triangular matrix appear on the main diagonal. Verify this fact for the matrices in Prob- lems 43—45. 1 0 3 7 43. [b i} 44. ["3 we] 45. [a a a] o o 2 46. Use properties of determinants (Sec. 3 4) to demonstrate why the diagonal eigenvalue property hoitls in general for triangular matriCes. Eigenvalues of a Transpose For Problems 47—49, let A be a square matrix and detemzine tbefollowing facts about its transpose. 47. Show that A is invortible if and only if'AI is invertible. 48. Show that A and AT have the same eigenvalues. 49. Give an example of matrices A and AT to show that the corresponding eigenvectors for a given it are not the same 50. Orthogonal Eigenvectors Let A be a symmetric matrix (that is, A = A?) with distinct eigenvalues Ag and Ag. For such a matrix, if v, and a; are eigenvectors belonging to the distinct eigenvaiues A] and A2, respectively, then v. and it; are orthogonal. {a} Illustrate this for A E 2 — [2 il (b) Prove fact for an n x a symmetric matrix. Use the fact that it; . a; M iii; (as a matrix product). 51. Another Eigenspace Find the eigenvalues, if any, and the corresponding eigenspaces for the linear transfomta~ tion "I : ii”; —> ll”; defined by Nat:2 + bx + c) m bx + c. 52. Checking Up on Eigenvalues In a quadratic equation with leading coefficient 1, the negative of the coefficient of the linear term is the sum of the roots, and the constant term is the product of the roots. in) i’rove these properties by expanding the factored quadratic (x ”lull-1‘ ~ 9&2) = 0 (b) Compare this result to equation (5). Explain how to tie [ermine from a matrix, without solving the character* istic equation, the sum and product of its eigenvalues. (c) Illustrate these results for the matrix l3 3i Looking for Matrices For Problems 53—57, find all the 2 x 2 matrices with the desired properties _ O . . 53. v = [1 ES an eigenvector. , a 1 . . :4. v = [l] is an eigenvector 7 55. [ a] and [“1 ] are eigenvectors, with double eigenvalue A. = 1 56, [a] and [Mi] are eigenvectors, with eigenvalues I and 2, respectively. ~1 2 with the same eigenvalue A = ~—i. 57. [ I] and {0] are eigenvectors, Linear Transformations in the Plane For Problems 58—62, find the eigenvalues, if any, and corresponding eigenvectors for the transformations in Table 5.1.1 58. Reflections about the .rnaxis. 59. Reflections about the y—mtis. 60. Clockwise rotation of tit/4 about the origin. anus. m4 “saris"! a .326 Chapter 5 Linear iransformations 61. Reflection about the line y z x 62. Shear of 2 in the y-direction. Cayley-Hamiiton We have met these nineteenth-century mathematicians before: Cayley in Sec 3.1 and Hamilton in Sec. 35. They proved the following theorem Cayley~Hamilton Theorem .. . . _ A matrix satisfies its own characteristic equation. U12 + hit a- c = 0 is the characteristic equation of the 2 x '2 matrix A, for example, then A2 + bA 4— cI = 0 Verify this for each matrix in Problems 6.3»66. 1 r a r 63. [4 1] e4. [*1 0] 1 i O 1 l 2 65. G 1 1 66. O 2 3 V D 0 1 I 0 4 Inverses by Cayley-Harnilton For an invertible 3 x 3 ma» his A, we can write, using the Cayley-Hamilton Theorem, A3 + hA“ wlw cA + dI ~— 0 where b c and cl are coefi’icients of the characteristic equation of A If we multiply through on the left by A“ ,we get A2 + hA + Cl + dA“ .— -l), which can be solved for A““‘ Use this method to calculate the inverse: of Problems 67 and 68. 2 0 O i .2 mi 67. l —i <3] 68. [ 1 0 1] [ml 0 i 4 W4 5 69. Develop a Cayley-Hamiiton lurmula for the inverse of a 2 x .2 matrix, and apply it to compute the inverses of the following. t: a (b) [j j] 70. Trace and Determinant as Parameters Express the eigenvalues of a 2 x 2 matrix in terms of its trace and its determinant, 71. Raising the Order Generalize the results of Problem 70 to the characteristic equation and eigenvalues of a 3 x 3 matrix Then illustrate these results for the matrix 1 2&1 101 4—4 5 Eigenvalues and Conversion Using the method of Sec. 4. 7, convert each (lifi'erential equation in Problems 72—75 to a system of first-order equations. Then verify that the charac» ter‘istic roots of the DE are the same as the eigenvalues of the matrix of the converted linear system 72. y” w 3” ~— Zy = O 73. y” - 23” + 5r = 0 74. 3”” + 2y" — y’ — .2312 0 75; it” ~ 2y” - 5y’ «r 6r = 0 Eigenfunction Boundary-Value Problems For what values of the normegative constant it in the equation y” + ity = 0 do there exist nonzero solutions satisfying the boundary condi» tions in Problems 7’6w78 ? The values of it are called eigenvalw ues and the corresponding solutions are called eigenfitnctions. 76. y(0) = 0, yin) : 0 77. y’(0) = 0, fit?) = 78. y(—n) = y(n'), «VT-H} a VG?) 79. Computer Lab: Eigenvcctors For each matrix (a)-(h), find the eigenvalues and eigenvectors. To make quick work of this, use computer software leg, IDE, Derive, Matlab, or other computer algebra systems}. From your results, iist coniectares (and illustrations) of what you might be able to predict for eigenvalues and eigenvectors from just looking at a 2 x 2 matrix (without calcuiations). Eigen~Engine For 2 x 2 matrices you can see the eigen— vectors as well as their coordinate values. (a) ll) (1)] {b} [5 2] (Cl [3 ii “1) [i ii (at ii ll (fl Li i] (g [3 i] ““0 lo 3] 80. Suggested Journal Entry Suppose that you had calcu- iated the first few powers of a matrix A How could you use the Cayley-Hamilton Theorem (in the introduction to Problems 63—66) to compute higher powers of A without doing any further matrix multiplications? Could a similar scheme be used to find powers of A‘1 for invertible A? ...
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