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Chapter 7: Eigenvalues and Eigenvectors

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Linear Algebra With Applications
Pages: 310 - 384
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Illustration

344 Questions for Chapter 7: Eigenvalues and Eigenvectors

  1. Prove the fundamental theorem of algebra for cubic polynomials with real coefficients.

    Found on Page 371

  2. Find all2×2matrices for which[12]is an eigenvector with associated eigenvalue 5

    Found on Page 323

  3. For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

    Found on Page 345

  4. Use eigenvalues to determine the stability of a dynamical systemx→(t+1)=Ax→(t). Analyse the dynamical system, where A is a real2X2matrix with eigenvaluesp±iqFor the matrices A in Exercises 1 through 10, determine whether the zero state is a stable equilibrium of the dynamicalsystem.x→(t+1)=Ax→(t)

    Found on Page 380

  5. There exists a real 5 × 5 matrix without any real eigenvalues.

    Found on Page 383

  6. For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

    Found on Page 336

  7. Express the polynomial f(λ)=λ3-3λ2+7λ-5 as a product of linear factors over ℝ.

    Found on Page 371

  8. For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

    Found on Page 345

  9. For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

    Found on Page 336

  10. Consider the matrices A in Exercises 11 through 16. For which real numbers k is the zero state a stable equilibrium of the dynamical systemx→(t+1)=Ax→(t)?

    Found on Page 380

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