Problem #4: Consider the following matrix: $A = \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix}$ (a) Review §8.7 of the text. (b) Find all of the eigenvalues of A. (c) Find the eigenvectors corresponding to the eigenvalues you found in (b). [Caution: The eigenvalues and eigenvectors are complex here! This is an important case in upcoming problems in Chapter 10.]
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Step 1: (b) To find the eigenvalues of A, we need to solve the characteristic equation, which is given by $\det(A - \lambda I) = 0$, where $\lambda$ represents the eigenvalues and $I$ is the identity matrix. Show more…
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In both parts of this problem, consider the matrix \[ A=\left[\begin{array}{ll} 1 & 2 \\ 4 & 3 \end{array}\right] \], with eigenvalues $\lambda_{1}=5$ and $\lambda_{2}=-1 .$ See Example 1. a. Are the column vectors of the matrices $A-\lambda_{1} I_{2}$ and $A-\lambda_{2} I_{2}$ eigenvectors of $A ?$ Explain. Does this work for other $2 \times 2$ matrices? What about diagonalizable $n \times n$ matrices with two distinct eigenvalues, such as projections or reflections? Hint: Exercise 70 is helpful. b. Are the column vectors of \[ A-\left[\begin{array}{cc} \lambda_{1} & 0 \\ 0 & \lambda_{2} \end{array}\right] \] eigenvectors of $A ?$ Explain.
Eigenvalues and Eigenvectors
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(a) Show that the matrix $\mathbf{A}=\left[ \begin{array}{ll}{1} & {-1} \\ {4} & {-3}\end{array}\right]$ has the repeated eigenvalue $r=-1$ and that all the eigenvectors are of the form $\mathbf{u}=s \operatorname{col}(1,2)$ (b) Use the result of part (a) to obtain a nontrivial solu- $\quad$ tion $\mathbf{x}_{1}(t)$ to the system $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$ (c) To obtain a second linearly independent solution to $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$ try $\mathbf{x}_{2}(t)=t e^{-t} \mathbf{u}_{1}+e^{-t} \mathbf{u}_{2}$ [Hint: Substitute $\mathbf{x}_{2}$ into the system $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$ and derive the relations $$ (\mathbf{A}+\mathbf{I}) \mathbf{u}_{1}=\mathbf{0}, \quad(\mathbf{A}+\mathbf{I}) \mathbf{u}_{2}=\mathbf{u}_{1} $$ since $\mathbf{u}_{1}$ must be an eigenvector set $\mathbf{u}_{1}=$ $\operatorname{col}(1,2)$ and solve for $\mathbf{u}_{2} . ]$ (d) What is $(\mathbf{A}+\mathbf{I})^{2} \mathbf{u}_{2} ?$ (In Section $9.8, \mathbf{u}_{2}$ will be identified as a generalized eigenvector.)
Matrix Methods for Linear Systems
Homogeneous Linear Systems with Constant Coefficients
PROBLEM 85: Let A be the matrix A = [ 0 1 0; -4 4 0; -2 0 1 ] (a) Find the eigenvalues and eigenvectors of A. (b) Find the algebraic multiplicity a_lambda, and the geometric multiplicity, g_lambda of each eigenvalue. (c) For one of the eigenvalues you should have g_lambda < a_lambda. (If not, redo the preceding parts!) Find a generalized eigenvector for this eigenvalue. (d) Verify that the eigenvectors and generalized eigenvectors are all linearly independent. (e) Find a fundamental set of solutions for x' = Ax.
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