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Chapter 5: Eigenvalues and Eigenvectors

Expert-verified
Linear Algebra and its Applications
Pages: 267 - 330
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Illustration

265 Questions for Chapter 5: Eigenvalues and Eigenvectors

  1. In Exercises 9–16, find a basis for the eigenspace corresponding to each listed eigenvalue.

    Found on Page 267

  2. (M) Exercises 7-12 require MATLAB or other computational aid. In Exercises 7 and 8, use the power method with the \({{\bf{x}}_0}\) given. List \(\left\{ {{{\bf{x}}_k}} \right\}\) and \(\left\{ {{\mu _k}} \right\}\) for \(k = 1, \ldots .5.\) In Exercises 9 and 10, list \({\mu _5}\) and \({\mu _6}\).

    Found on Page 267

  3. Question: Exercises 9–14 require techniques from Section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for \(3 \times 3\)determinants described prior to Exercises 15–18 in Section 3.1. (Note:Finding the characteristic polynomial of a \(3 \times 3\)matrix is not easy to do with just row operations, because the variable \(\lambda \)is involved.)

    Found on Page 267

  4. In Exercises 9 and 18,construct the general solution of\(x' = Ax\)involving complex Eigen functions and then obtain the general real solution. Describe the shapes of typical trajectories.

    Found on Page 267

  5. Show that if \(A\) is diagonalizable, with all eigenvalues less than 1 in magnitude, then \({A^k}\) tends to the zero matrix as \(k \to \infty \). (Hint: Consider \({A^k}x\) where \(x\) represents any one of the columns of \(I\).)

    Found on Page 267

  6. Another estimate can be made for an eigenvalue when an approximate eigenvector is available. Observe that if \(A{\bf{x}} = \lambda {\bf{x}}\), then \({{\bf{x}}^T}A{\bf{x}} = {{\bf{x}}^T}\left( {\lambda {\bf{x}}} \right) = \lambda \left( {{{\bf{x}}^T}{\bf{x}}} \right)\) and the Rayleigh quotient

    Found on Page 267

  7. Question: Exercises 9-14 require techniques section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for \(3 \times 3\) determinants described prior to Exercise 15-18 in Section 3.1. (Note: Finding the characteristic polynomial of a \(3 \times 3\) matrix is not easy to do with just row operations, because the variable \(\lambda \) is involved.)

    Found on Page 267

  8. In Exercises 9–16, find a basis for the eigenspace corresponding to each listed eigenvalue.

    Found on Page 267

  9. In Exercises 9 and 18,construct the general solution of\(x' = Ax\)involving complex Eigen functions and then obtain the general real solution. Describe the shapes of typical trajectories.

    Found on Page 267

  10. In Exercises 11 and 12, mark each statement True or False. Justify each answer.

    Found on Page 267

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