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Understanding Eigenvalues and Eigenvectors in Linear Algebra

This course explores the fundamentals of linear algebra with a focus on eigenvalues and eigenvectors. We delve into characteristic polynomials and their significance. Expect to analyze transformation matrices and understand how they interact with eigenvectors. Assignments include solving specific problems from the textbook, "Linear Algebra With Applications" by Otto Bretscher. The goal is to equip students with the tools to determine the behavior of matrices through their eigenvalues.

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Understanding Eigenvalues and Eigenvectors in Linear Algebra

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  1. Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, Second Edition Otto Bretscher

  2. Friday, April 18 Chapter 7.2 • Page 310 Problems 6,8,10,20 • Main Idea: How do you tell what a matrix is going to do? • Key Words: Eigen Value, Eigen Vector, Characteristic Polynomial • Goal: Introduction to eigenvalues and eigen vectors.

  3. Previous Assignment. • Page 300 Problem 2 • Let A be an invertible nxn matrix • and V an eigenvector of A with associated eigen value c • If V is an eigenvector of A^(-1) ? If So, what is its • eigenvalue. • If A stretches V by a factor of c, then A^(-1) must • shrink V by a factor of 1/c.

  4. Page 300 Problem 4 • Let A be an invertible nxn matrix and V an eigenvector • of A with associated eigen value c • Is V an eigen vector of 7A? IF so, what is the eigenvalue? • If A streches V by a factor of c, then 7 A stretches • V by a factor of 7 c.

  5. Page 300 Problem 6 • If a vector V is an eigenvector of both A and B, is • V necessarily an eigen vector of AB? • Let A V = a V and B V = b V. • A B V = A b V = b A V = ba V • V is an eigen vector of AB and the eigenvalue is ab.

  6. Page 300 Problem 10 Find all 2x2 matrices for which | 1 | | 2 | is an eigen vector for eigen value 5 • | a b | | 1 | = | 5 | • | c d | | 2 | |10| • a+2b = 5 • c+2d = 10

  7. a b c d • 1 2 0 0 5 • 0 0 1 2 10 • | a | | -2| | 0 | | 5 | • | b | = b | 1| + d | 0 | + | 0 | • | c | | 0| |-2 | | 10 | • | d | | 0| | 1 | | 0 | • | -2 b + 5 b | • | -2 d +10 d |

  8. Check. • | -2 b + 5 b || 1 | | 5| • | -2 d +10 d || 2 | = |10 |

  9. Page 300 Problem 40 • Suppose that V is an eigenvector of the nxn • matrix A, with eigen value 4. Explain why • V is an eigenvector of A^2 + 2A + 3 In. • What is its associated eigenvalue. • (A^2 + 2 A + 3I)V = A(AV) + 2 AV + 3 V • = (16+8+3)V • = 27 V.

  10. Find the Eigen values and vectors of • | 2 -1 -1 | • |-1 2 -1 | • |-1 -1 2 |

  11. | 2-x -1 -1 | • Det[A-xI = | -1 2-x -1 | • | -1 -1 2-x | • | 2-x -1 -1 | • Det[A-xI = |-3+x 3-x 0 | • | -1 -1 2-x |

  12. | 2-x -1 -1 | • Det[A-xI = | -1 2-x -1 | • | -1 -1 2-x | • | 2-x -1 -1 | • Det[A-xI = |-3+x 3-x 0 | • | -1 -1 2-x |

  13. | 2-x -1 -1 | • Det[A-xI =(x-3) | 1 -1 0 | • | -1 -1 2-x | • | 2-x -1 -1 | • Det[A-xI =(x-3) |-1+x 0 1 | • |-3+x 0 3-x |

  14. Det[A-xI =(x-3) |-1+x 1 | • |-3+x 3-x | • Det[A-xI =(x-3)^2 |-1+x 1 | • | 1 -1 | • Det[A-xI =(x-3)^2 (-x)

  15. The eigen values are 3,3,0 • x=3 A-3I = | -1 -1 -1 | • | -1 -1 -1 | • | -1 -1 -1 | • RCF(A-3I) = | 1 1 1 | • | 0 0 0 | • | 0 0 0 | • [V1 V2 ] = | -1 -1 | • | 1 0 | • | 0 1 |

  16. Check: • | 2 -1 -1 | | -1 -1 | | -3 -3 | | -1 -1 | • |-1 2 -1 | | 1 0 | = | 3 0 | = 3| 1 0 | • |-1 -1 2 | | 0 1 | | 0 3 | | 0 1 |

  17. x = 0 • | 2 -1 -1 | | 1 1 -2 | | 1 0 -1 | • |-1 2 -1 | ~ | 0 -3 3 | ~ | 0 1 -1 | • |-1 -1 2 | | 0 3 -3 | | 0 0 0 | • | 1 | • V3 = | 1 | • | 1 |

  18. Check: • | 2 -1 -1 | | 1 | | 0 | | 1 | • |-1 2 -1 | | 1 | = | 0 | = 0 | 1 | • |-1 -1 2 | | 1 | | 0 | | 1 |

  19. Diagonalization: • -1 • | -1 -1 1 | | 2 -1 -1 | | -1 -1 1 | • | 1 0 1 | | -1 2 -1 | | 1 0 1 | • | 0 1 1 | | -1 -1 2 | | 0 1 1 |

  20. | -1 2 -1 | | 2 -1 -1 | | -1 -1 1 | • 1/3 | -1 -1 2 | | -1 2 -1 | | 1 0 1 | • | 1 1 1 | | -1 -1 2 | | 0 1 1 |

  21. | -1 2 -1 | | -1 -1 1 | • | -1 -1 2 | | 1 0 1 | • | 0 0 0 | | 0 1 1 | • | 3 0 0 | • | 0 3 0 | • | 0 0 3 |

  22. Find a formula for the Fibonacci Numbers. • fo = 1 • f1 = 1 • f2 = 2 • f3 = 3 • fn = fn-1+fn-2.

  23. | 0 1 | | fn | = | fn+1 | = | fn+1 | • | 1 1 | | fn+1 | | fn+fn+1| | fn+2 | • n • F | 1 | = | fn | • | 1 | | fn+1 |

  24. Det[ F-xI ] = | -x 1 | = x^2 - x - 1 • | 1 1-x |

  25. Let the polynomial factor into (x-a)(x-b) where • 1+Sqrt[5] • a = ----------- • 2 • 1-Sqrt[5] • b = ---------- • 2

  26. There exist matrices P such that P^(-1) F P = | a 0 | | 0 b | • F = P | a 0 | P^(-1) • | 0 b | • F^n = P | a^n 0 | P^(-1) • | 0 b^n |

  27. | fn | = P | a^n 0 | P^(-1) • | fn+1 | | 0 b^n |

  28. So we have to compute P. • | -a 1 | ~ | 1 1-a | • | 1 1-a | | 0 0 | • | -b 1 | ~ | 1 1-b | • | 1 1-b | | 0 0 |

  29. P = [V1 V2] = |-1+a -1+b | = |-b -a | • | 1 1 | | 1 1 | • P^(-1) = 1/(a-b) | 1 a | • | -1 -b |

  30. F^n = 1/(a-b) | -b -a | | a^n 0 | | 1 a | • | 1 1 | | 0 b^n | | -1 -b |

  31. | n n n n | • | -(a b) + a b a b (a - b ) | • | -------------- -(-------------) | • | a - b a - b • | • | n n 1 + n 1 + n | • | a - b a - b | • | ------- --------------- | • | a - b a - b |

  32. F^n | 1 | = | fn | • | 1 | | fn+1 | • | n 1 + n n | • | -(a b) - a b + a b (1 + b) | • | --------------------------------- | • | a - b | • | | • | n 1 + n n | • | a + a - b (1 + b) | • | ------------------------ | • | a - b |

  33. So fn = -a^n b - a^(1+n) b + a b^n (1+b) • ----------------------------------------- • a-b

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