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Numerical Methods & Linear Algebra

Math 2890-003

Spring 2016 Homework

Chapter 5 — Due Apr 26

  1. (1 point) Let \[A=\left(\begin{array}{rrrr} 0 & 48 & -38 & -8 \\ 12 & -43 & 24 & 12 \\ 14 & -48 & 25 & 14 \\ 19 & -120 & 82 & 27 \end{array}\right)\quad\text{and}\quad \lambda = 8.\] Find an eigenvector for the matrix $A$ that corresponds to the given eigenvalue $\lambda$. Show and explain your work.
  2. (1 point) Let \[A=\left(\begin{array}{rrrr} 72 & -5 & 5 & -28 \\ -20 & -9 & -7 & -4 \\ -196 & 20 & -4 & 98 \\ 182 & -10 & 17 & -61 \end{array}\right)\quad\text{and}\quad x = \left(\begin{array}{r} 1 \\ 0 \\ -4 \\ 2 \end{array}\right).\] Find the eigenvalue for the matrix $A$ that corresponds to the given eigenvector $x.$ Show and explain your work.
  3. (1 point) Let \[A=\left(\begin{array}{rr} -17 & 7 \\ -42 & 18 \end{array}\right).\] Find the eigenvalues (including multiplicities) of $A$. Show and explain your work.
  4. (1 point) Let \[A=\left(\begin{array}{rrr} -1 & 6 & 18 \\ 0 & -4 & -21 \\ 0 & 0 & 3 \end{array}\right).\] Find the eigenvalues (including multiplicities) of $A$. Show and explain your work.
  5. (1 point) Let \[A=\left(\begin{array}{rrrr} 13 & -9 & 0 & 0 \\ 18 & -14 & 0 & 0 \\ 10 & -6 & -8 & -2 \\ -2 & 6 & 12 & 2 \end{array}\right).\] Find the eigenvalues (including multiplicities) of $A$. Show and explain your work.
  6. (1 point) Let \[A=\left(\begin{array}{rrr} -3 & 0 & 0 \\ 6 & -5 & 0 \\ -12 & 4 & -1 \end{array}\right).\] Find an invertible matrix $P$ and a diagonal matrix $D$ such that $AP=PD$, or explain why no such matrices exist.
  7. (1 point) Let \[A=\left(\begin{array}{rrrr} 5 & 3 & 6 & -24 \\ 0 & 4 & 0 & -18 \\ 0 & 0 & 4 & 6 \\ 0 & 0 & 0 & -2 \end{array}\right).\] Find an invertible matrix $P$ and a diagonal matrix $D$ such that $AP=PD$, or explain why no such matrices exist.
  8. (1 point) Let \[A=\left(\begin{array}{rrr} -3 & 5 & -3 \\ 0 & -3 & 0 \\ 0 & 0 & -4 \end{array}\right).\] Find an invertible matrix $P$ and a diagonal matrix $D$ such that $AP=PD$, or explain why no such matrices exist.
  9. (1 point) Let \[A=\left(\begin{array}{rrrr} 1 & 1 & 19 & 9 \\ -2 & -2 & -31 & -13 \\ 0 & 0 & -7 & -2 \\ 0 & 0 & 12 & 3 \end{array}\right).\] Find an invertible matrix $P$ and a diagonal matrix $D$ such that $AP=PD$, or explain why no such matrices exist.
  10. (1 point) Let $A=PDP^{-1}$ where \[P=\left(\begin{array}{rrrr} 1 & 3 & -1 & 0 \\ -2 & -5 & -2 & 1 \\ 2 & 9 & -13 & 1 \\ 4 & 13 & -7 & 0 \end{array}\right) \quad\text{and}\quad D=\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right). \] Then \[P^{-1}=\left(\begin{array}{rrrr} 45 & -8 & 8 & -19 \\ -16 & 3 & -3 & 7 \\ -4 & 1 & -1 & 2 \\ 2 & 2 & -1 & 1 \end{array}\right) \quad\text{and}\quad A=\left(\begin{array}{rrrr} -91 & 17 & -17 & 40 \\ 178 & -21 & 27 & -69 \\ -226 & 51 & -45 & 105 \\ -388 & 71 & -71 & 169 \end{array}\right). \] Find the eigenvalues of $A$ and for each eigenvalue find a corresponding eigenvector.
  11. (1 point) Let \[x_1=\left(\begin{array}{r} 9 \\ -2 \\ 5 \\ -1 \end{array}\right)\ x_2=\left(\begin{array}{r} -4 \\ -6 \\ 1 \\ -5 \end{array}\right)\ x_3=\left(\begin{array}{r} -6 \\ -1 \\ 8 \\ 7 \end{array}\right)\ x_4=\left(\begin{array}{r} -1 \\ 8 \\ 0 \\ 3 \end{array}\right)\] and \[\lambda_1=-5,\ \lambda_2=1,\ \lambda_3=-8,\ \lambda_4=7.\] Write down a matrix $A$ that has the given vectors as eigenvectors with the corresponding scalars as the eigenvalues.
  12. (1 point) Let \[A=\left(\begin{array}{rrr} 0.2 & -10.8 & 2.7 \\ 1.1 & 13.7 & -3.1 \\ 4.4 & 50.4 & -11.3 \end{array}\right).\] Is the origin an attractor, repeller or saddle point for the discrete dynamical system $x_{k+1} = A x_k$? How do you know? Hint: It may help to know that $AP=P D$ where \[P=\left(\begin{array}{rrr} 1 & 0 & 3 \\ -1 & 1 & -1 \\ -4 & 4 & -3 \end{array}\right)\quad\text{and}\quad D = \left(\begin{array}{rrr} 0.2 & 0 & 0 \\ 0 & 1.3 & 0 \\ 0 & 0 & 1.1 \end{array}\right).\]
  13. (1 point) Let \[A=\left(\begin{array}{rrrr} -56.1 & -1.6 & -13.6 & -6 \\ -115.2 & -2.5 & -27.6 & -12 \\ 233.1 & 8.8 & 57.7 & 24.3 \\ 53.1 & -3.2 & 10.2 & 7 \end{array}\right).\] Is the origin an attractor, repeller or saddle point for the discrete dynamical system $x_{k+1} = A x_k$? How do you know? Hint: It may help to know that $AP=P D$ where \[P=\left(\begin{array}{rrrr} 1 & 4 & -4 & -4 \\ 2 & 9 & -12 & -9 \\ -4 & -18 & 25 & 19 \\ -1 & 0 & -15 & -2 \end{array}\right)\quad\text{and}\quad D = \left(\begin{array}{rrrr} 1.1 & 0 & 0 & 0 \\ 0 & 1.5 & 0 & 0 \\ 0 & 0 & 1.6 & 0 \\ 0 & 0 & 0 & 1.9 \end{array}\right).\]
  14. (1 point) Let \[A=\left(\begin{array}{rrr} 12.3 & 2.4 & -1.1 \\ -30.8 & -5.8 & 2.8 \\ 59.4 & 12 & -5.2 \end{array}\right).\] Is the origin an attractor, repeller or saddle point for the discrete dynamical system $x_{k+1} = A x_k$? How do you know? Hint: It may help to know that $AP=P D$ where \[P=\left(\begin{array}{rrr} 1 & 2 & 1 \\ -4 & -7 & 0 \\ 2 & 6 & 11 \end{array}\right)\quad\text{and}\quad D = \left(\begin{array}{rrr} 0.5 & 0 & 0 \\ 0 & 0.6 & 0 \\ 0 & 0 & 0.2 \end{array}\right).\]
  15. (1 point) Let \[A=\left(\begin{array}{rrr} 28 & -6 & -42 \\ -22 & 4 & 34 \\ 26 & -6 & -40 \end{array}\right).\] Is the origin an attractor, repeller or saddle point for the differential equation $y' = A y$? How do you know? Hint: It may help to know that $AP=P D$ where \[P=\left(\begin{array}{rrr} 1 & -1 & -3 \\ -1 & 2 & 1 \\ 1 & -1 & -2 \end{array}\right)\quad\text{and}\quad D = \left(\begin{array}{rrr} -8 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 2 \end{array}\right).\]
  16. (1 point) Let \[A=\left(\begin{array}{rrrr} -47 & 24 & 9 & -3 \\ 69 & -51 & -16 & 5 \\ -393 & 244 & 83 & -29 \\ -81 & 52 & 19 & -13 \end{array}\right).\] Is the origin an attractor, repeller or saddle point for the differential equation $y' = A y$? How do you know? Hint: It may help to know that $AP=P D$ where \[P=\left(\begin{array}{rrrr} 1 & 2 & 1 & 0 \\ 2 & 5 & -2 & -1 \\ -2 & -7 & 11 & 4 \\ -3 & -7 & 3 & 4 \end{array}\right)\quad\text{and}\quad D = \left(\begin{array}{rrrr} -8 & 0 & 0 & 0 \\ 0 & -8 & 0 & 0 \\ 0 & 0 & -5 & 0 \\ 0 & 0 & 0 & -7 \end{array}\right).\]
  17. (1 point) Let \[A=\left(\begin{array}{rrr} 122 & -90 & -51 \\ 303 & -229 & -133 \\ -258 & 200 & 119 \end{array}\right).\] Is the origin an attractor, repeller or saddle point for the differential equation $y' = A y$? How do you know? Hint: It may help to know that $AP=P D$ where \[P=\left(\begin{array}{rrr} 1 & -3 & -3 \\ 3 & -8 & -5 \\ -3 & 7 & 2 \end{array}\right)\quad\text{and}\quad D = \left(\begin{array}{rrr} 5 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 6 \end{array}\right).\]
  18. (1 point) Suppose $AP=P D$ where \[P=\left(\begin{array}{rrrr} 1 & -1 & -3 & 1 \\ 4 & 1 & 2 & 0 \\ 0 & 5 & 0 & -3 \\ 5 & -2 & -1 & -4 \end{array}\right)\quad\text{and}\quad D =\left(\begin{array}{rrrr} -9 & 0 & 0 & 0 \\ 0 & -8 & 0 & 0 \\ 0 & 0 & -5 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right).\] Solve the discrete dynamical system $x_{k+1}=Ax_k$ where \[x_0=\left(\begin{array}{r} 8 \\ -19 \\ 10 \\ -46 \end{array}\right).\] Show your work.
  19. (1 point) Suppose $AP=P D$ where \[P=\left(\begin{array}{rrrr} 1 & -1 & -2 & 0 \\ 4 & 3 & -2 & 0 \\ 5 & -5 & 5 & -2 \\ -2 & 2 & -5 & -4 \end{array}\right)\quad\text{and}\quad D =\left(\begin{array}{rrrr} -4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right).\] Solve the initial value problem $y'=Ay$ where \[y(0)=\left(\begin{array}{r} -4 \\ -32 \\ 12 \\ -6 \end{array}\right).\] Show your work.
  20. (1 point) Let \[A=\left(\begin{array}{rrr} 7 & 24 & 16 \\ -16 & -43 & -26 \\ 16 & 39 & 22 \end{array}\right)\quad\text{and}\quad x_0=\left(\begin{array}{r} 1 \\ 1 \\ 1 \end{array}\right).\] Use the Power Method to find estimates $\mu_5$ and $x_5$ for the dominant eigenvalue of $A$ and its eigenvector. Give your answer either as rational numbers or decimals with at least four digits of accuracy.
  21. (1 point) Let \[A=\left(\begin{array}{rrr} 3.7 & 5.2 & 3.6 \\ -31.2 & 12.1 & -10.4 \\ -21.6 & -10.4 & -14.3 \end{array}\right),\quad x_0=\left(\begin{array}{r} 1 \\ 1 \\ 1 \end{array}\right)\quad\text{and}\quad \alpha=2.\] Use the Inverse Power Method to find the estimate $\nu_3$ for the eigenvalue of $A$ closest to $\alpha$ and the estimate $x_3$ for the corresponding eigenvector. Give your answer as decimals with at least four digits of accuracy.
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