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dy Find eigenvalues & eigenvectors. = [²:2] Y dt 3-2

dy Find eigenvalues & eigenvectors. = [²:2] Y dt 3-2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Understanding Eigenvalues and Eigenvectors** **Objective:** Learn how to find the eigenvalues and eigenvectors of a matrix. **Problem Statement:** Find the eigenvalues and eigenvectors for the following system: \[ \frac{d\vec{Y}}{dt} = \begin{bmatrix} -2 & -3 \\ 3 & -2 \end{bmatrix} \vec{Y} \] **Explanation:** In this problem, we are given a differential equation involving a matrix. Our task is to determine the eigenvalues and eigenvectors of the matrix: \[ \begin{bmatrix} -2 & -3 \\ 3 & -2 \end{bmatrix} \] **Steps to Solve:** 1. **Find the Characteristic Equation:** The characteristic equation is obtained by finding the determinant of the matrix subtracted by \( \lambda \) times the identity matrix. 2. **Solve for Eigenvalues:** Solve the characteristic equation for \( \lambda \). 3. **Determine Eigenvectors:** For each eigenvalue, substitute back into the equation \( (A - \lambda I) \vec{v} = 0 \) to find the corresponding eigenvector. By following these steps, one can find both the eigenvalues and eigenvectors for the given matrix.
Transcribed Image Text:**Title: Understanding Eigenvalues and Eigenvectors** **Objective:** Learn how to find the eigenvalues and eigenvectors of a matrix. **Problem Statement:** Find the eigenvalues and eigenvectors for the following system: \[ \frac{d\vec{Y}}{dt} = \begin{bmatrix} -2 & -3 \\ 3 & -2 \end{bmatrix} \vec{Y} \] **Explanation:** In this problem, we are given a differential equation involving a matrix. Our task is to determine the eigenvalues and eigenvectors of the matrix: \[ \begin{bmatrix} -2 & -3 \\ 3 & -2 \end{bmatrix} \] **Steps to Solve:** 1. **Find the Characteristic Equation:** The characteristic equation is obtained by finding the determinant of the matrix subtracted by \( \lambda \) times the identity matrix. 2. **Solve for Eigenvalues:** Solve the characteristic equation for \( \lambda \). 3. **Determine Eigenvectors:** For each eigenvalue, substitute back into the equation \( (A - \lambda I) \vec{v} = 0 \) to find the corresponding eigenvector. By following these steps, one can find both the eigenvalues and eigenvectors for the given matrix.
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