?Use your technology utility to duplicate the computations in Example 3

In Exercises 1-2, the distinct eigenvalues of a matrix are given. Determine whether \(A\) has a dominant eigenvalue, and if so, find it. a. \(\lambda_{1}=7, \lambda_{2}=3, \lambda_{3}=-8, \lambda_{4}=1\) b. \(\lambda_{1}=-5, \lambda_{2}=3, \lambda_{3}=2, \lambda_{4}=5\) Equation Transcription: Text Transcription: A lambda_1 = 7, lambda_2 = 3, lambda_3 = -8, lambda_4 = 1 lambda_1 = -5, lambda_2= 3, lambda_3 = 2, lambda_4 = 5

In Exercises 1-2, the distinct eigenvalues of a matrix are given. Determine whether \(A\) has a dominant eigenvalue, and if so, find it. a. \(\lambda_{1}=1, \lambda_{2}=0, \lambda_{3}=-3, \lambda_{4}=2\) b. \(\lambda_{1}=-3, \lambda_{2}=-2, \lambda_{3}=-1, \lambda_{4}=3\) Equation Transcription: Text Transcription: A lambda_1 = 1, lambda_2 = 0, lambda_3 = -3, lambda_4 = 2 lambda_1 = -3, lambda_2 = -2, lambda_3 = -1, lambda_4 = 3

In Exercises 3-4, apply the power method with Euclidean scaling to the matrix \(A\), starting with \(\mathbf{x}_{0}\) and stopping at \(\mathbf{x}_{4}\). Compare the resulting approximations to the exact values of the dominant eigenvalue and the corresponding unit eigenvector. \(\quad A=\left[\begin{array}{rr}5 & -1 \\ -1 & -1\end{array}\right] ; \quad \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 0\end{array}\right]\) Equation Transcription: ; Text Transcription: A x_0 x_4 A = [5 & -1 \\ -1 & -1] ; x_0 = [1 \\ 0]

In Exercises 3-4, apply the power method with Euclidean scaling to the matrix \(A\), starting with \(\mathbf{x}_{0}\) and stopping at \(\mathbf{x}_{4}\). Compare the resulting approximations to the exact values of the dominant eigenvalue and the corresponding unit eigenvector. \(A=\left[\begin{array}{rrr}7 & -2 & 0 \\ -2 & 6 & -2 \\ 0 & -2 & 5\end{array}\right] ; \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) Equation Transcription: []; [] Text Transcription: A x_0 x_4 A = [7 & -2 & 0 \\ -2 & 6 & -2 \\ 0 & -2 & 5] ; x_0 = [1 \\ 0 \\ 0]

In Exercises 5-6, apply the power method with maximum entry scaling to the matrix \(A\), starting with \(\mathbf{x}_{0}\) and stopping at \(\mathbf{x}_{4}\). Compare the resulting approximations to the exact values of the dominant eigenvalue and the corresponding scaled eigenvector. \(A=\left[\begin{array}{rr}1 & -3 \\ -3 & 5\end{array}\right] ; \quad \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) Equation Transcription: Text Transcription: A x_0 x_4 A = [1 & -3 \\ -3 & ] ; x_0 = [1 \\ 1]

In Exercises 5-6, apply the power method with maximum entry scaling to the matrix \(A\), starting with \(\mathbf{x}_{0}\) and stopping at \(\mathbf{x}_{4}\). Compare the resulting approximations to the exact values of the dominant eigenvalue and the corresponding scaled eigenvector. \(A=\left[\begin{array}{lll}3 & 2 & 2 \\ 2 & 2 & 0 \\ 2 & 0 & 4\end{array}\right] ; \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]\) Equation Transcription: []; [] Text Transcription: A x_0 x_4 A = [3 & 2 & 2 \\ 2 & 2 & 0 \\ 2 & 0 & 4] ; x_0 = [1 \\ 1 \\ 1]

Let \(A=\left[\begin{array}{rr}2 & -1 \\-1 & 2\end{array}\right] ; \quad \mathbf{x}_{0}=\left[\begin{array}{l}1 \\0\end{array}\right]\) a. Use the power method with maximum entry scaling to approximate a dominant eigenvector of \(A\) .Start with \(x_{0}\), round off all computations to three decimal places, and stop after three iterations. b. Use the result in part (a) and the Rayleigh quotient to approximate the dominant eigenvalue of \(A\). c. Find the exact values of the eigenvector and eigenvalue approximated in parts (a) and (b). d. Find the percentage error in the approximation of the dominant eigenvalue. Equation Transcription: Text Transcription: A = [2 & -1 \\-1 & 2 ; x_0 = [1 \\0] A x_0 A

Repeat the directions of Exercise 7 with \(A=\left[\begin{array}{llr}2 & 1 & 0 \\1 & 2 & 0 \\0 & 0 & 10\end{array}\right] ; \mathbf{x}_{0}=\left[\begin{array}{l}1 \\1 \\1\end{array}\right]\) Equation Transcription: []; [] Text Transcription: A = [2 & 1 & 0 \\1 & 2 & 0 \\0 & 0 & 10] ; x_0 = [1 \\1 \\1]

In Exercises 9-10, a matrix \(A\) with a dominant eigenvalue and a sequence \(\mathbf{x}_{0}, A \mathbf{x}_{0}, \ldots, A^{5} \mathbf{x}_{0}\) are given. Use Formulas (9) and (10) to approximate the dominant eigenvalue and a corresponding eigenvector. \(\quad A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right] ; \quad \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 0\end{array}\right], A \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 2\end{array}\right], A^{2} \mathbf{x}_{0}=\left[\begin{array}{l}5 \\ 4\end{array}\right]\) \(A^{3} \mathbf{x}_{0}=\left[\begin{array}{l}13 \\14\end{array}\right], A^{4} \mathbf{x}_{0}=\left[\begin{array}{l}41 \\40\end{array}\right], A^{5} \mathbf{x}_{0}=\left[\begin{array}{l}121 \\122\end{array}\right]\) Equation Transcription: , , , , Text Transcription: A x_0, Ax_0, ..., A^5{x}_{0} A = [1 & 2 \\ 2 & 1] ; x_0 = [1 \\ 0], Ax_0 = [1 \\ 2], A^2{x}_{0} = [5 \\ 4] A^3{x}_{0} = [13 \\14], A^4{x}_{0} = [41 \\40], A^5 {x}_{0} = [121 \\122]

In Exercises 9-10, a matrix \(A\) with a dominant eigenvalue and a sequence \(\mathbf{x}_{0}, A \mathbf{x}_{0}, \ldots, A^{5} \mathbf{x}_{0}\) are given. Use Formulas (9) and (10) to approximate the dominant eigenvalue and a corresponding eigenvector. \(A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right] ; \quad \mathbf{x}_{0}=\left[\begin{array}{l}0 \\ 1\end{array}\right], A \mathbf{x}_{0}=\left[\begin{array}{l}2 \\ 1\end{array}\right], A^{2} \mathbf{x}_{0}=\left[\begin{array}{l}4 \\ 5\end{array}\right]\) \(A^{3} \mathbf{x}_{0}=\left[\begin{array}{l}14 \\13\end{array}\right], A^{4}\mathbf{x}_{0}=\left[\begin{array}{l}40 \\41\end{array}\right], A^{5}\mathbf{x}_{0}=\left[\begin{array}{l}122 \\121\end{array}\right]\) Equation Transcription: , , , , Text Transcription: A x_0, Ax_0, ..., A^5{x}_{0} A = [1 & 2 \\ 2 & 1] ; x_0 = [0 \\ 1], Ax_0 = [2 \\ 1], A^2{x}_{0} = [4 \\ 5] A^3{x}_{0} = [14 \\13], A^4{x}_{0} = [40 \\41], A^5 {x}_{0} = [122 \\121]

Consider matrices \(A=\left[\begin{array}{rr}-1 & 0 \\0 & 0\end{array}\right] \text { and } \mathbf{x}_{0}=\left[\begin{array}{l}a \\b\end{array}\right]\) where \(\mathrm{x}_{0}\) is a unit vector and \(a \neq 0\). Show that even though the matrix \(A\) is symmetric and has a dominant eigenvalue, the power sequence (1) in Theorem 9.2.1 does not converge. This shows that the requirement in that theorem that the dominant eigenvalue be positive is essential. Equation Transcription: and Text Transcription: A = [-1 & 0 \\0 & 0] and x_0 = [a \\b] x_0 a neq 0 A

Use the power method with Euclidean scaling to approximate the dominant eigenvalue and a corresponding eigenvector of \(A\). Choose your own starting vector, and stop when the estimated percentage error in the eigenvalue approximation is less than 0.1% a. \(\left[\begin{array}{rrr}1 & 3 & 3 \\ 3 & 4 & -1 \\ 3 & -1 & 10\end{array}\right]\) b. \(\left[\begin{array}{rrrr}1 & 0 & 1 & 1 \\ 0 & 2 & -1 & 1 \\ 1 & -1 & 4 & 1 \\ 1 & 1 & 1 & 8\end{array}\right]\) Equation Transcription: []; [ ] Text Transcription: A [1 & 3 & 3 \\ 3 & 4 & -1 \\ 3 & -1 & 10] [1 & 0 & 1 & 1 \\ 0 & 2 & -1 & 1 \\ 1 & -1 & 4 & 1 \\ 1 & 1 & 1 & 8]

Repeat Exercise 12, but this time stop when all corresponding entries in two successive eigenvector approximations differ by less than 0.01 in absolute value.

Repeat Exercise 12 using maximum entry scaling.

Prove: If \(A\) is a nonzero \(n \times n\) matrix, then \(A^{T} A$ and $A A^{T}\) have positive dominant eigenvalues. Equation Transcription: Text Transcription: A n x n A^T A AA^T

(For readers familiar with proof by induction) Let \(A\) be an \(n \times n\) matrix, let \(\mathrm{x}_{0}\) be a unit vector in \(R^{n}\), and define the sequence \(\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}, \ldots\) by \(\mathbf{x}_{1}=\frac{A \mathbf{x}_{0}}{\left\|A \mathbf{x}_{0}\right\|}, \quad \mathbf{x}_{2}=\frac{A \mathbf{x}_{1}}{\left\|A \mathbf{x}_{1}\right\|}, \ldots, \quad \mathbf{x}_{k}=\frac{A \mathbf{x}_{k-1}}{\left\|A \mathbf{x}_{k-1}\right\|}, \ldots\) Prove by induction that \(\mathbf{x}_{k}=A^{k} \mathbf{x}_{0} /\left\|A^{k} \mathbf{x}_{0}\right\|\). Equation Transcription: Text Transcription: A n x n x_0 R^n x_1, x_2, ...,x}_k, ... x_1 = frac{Ax_0}{||Ax_0||}, x_2 = frac{Ax_1}{||Ax_1||}, ..., x_k = frac{Ax_k-1}{||Ax_k-1||}, ... x_k = A^k{x_0} /||A^k{x_0}||

(For readers familiar with proof by induction) Let \(A\) be an \(n \times n\) matrix, let \(\mathrm{x}_{0}\) be a nonzero vector in \(R^{n}\), and define the sequence \(\mathbf{x}_{1},\mathbf{x}_{2}, \ldots, \mathbf{x}_{k}, \ldots\) by \(\begin{array}{r}\mathbf{x}_{1}=\frac{A \mathbf{x}_{0}}{\max \left(A \mathbf{x}_{0}\right)}, \quad \mathbf{x}_{2}=\frac{A \mathbf{x}_{1}}{\max \left(A \mathbf{x}_{1}\right)}, \ldots \\\mathbf{x}_{k}=\frac{A \mathbf{x}_{k-1}}{\max \left(A \mathbf{x}_{k-1}\right)}, \ldots\end{array}\) Prove by induction that \(\mathbf{x}_{k}=\frac{A^{k} \mathbf{x}_{0}}{\max \left(A^{k} \mathbf{x}_{0}\right)}\) Equation Transcription: Text Transcription: A n x n x_0 R^n x_1, x_2, ...,x}_k, ... x_1 = frac{Ax_0}{m (Ax_0)}, x_2 = frac{Ax_1}{m (Ax_1)}, ..., x_k = frac{Ax_k-1}{m (Ax_k-1)}, ... x_k = frac {A^k{x_0}{m (A^k{x_0)}

Use your technology utility to duplicate the computations in Example 2.

Use your technology utility to duplicate the computations in Example 3.