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2) Suppose that A is a 3 × 3 matrix for the reflection across the plane 2х + Зу — 52 problem can be done without any computations, just think about the definition of eigenvalues and eigenvectors, and the imagine what happens in a reflection. 0. What are the eigenvalues of A? Explain your answer. This

2) Suppose that A is a 3 × 3 matrix for the reflection across the plane 2х + Зу — 52 problem can be done without any computations, just think about the definition of eigenvalues and eigenvectors, and the imagine what happens in a reflection. 0. What are the eigenvalues of A? Explain your answer. This

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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### Understanding Eigenvalues in the Context of a Reflection Matrix **Question 2:** Suppose that \( A \) is a \( 3 \times 3 \) matrix for the reflection across the plane \( 2x + 3y - 5z = 0 \). What are the eigenvalues of \( A \)? Explain your answer. This problem can be done without any computations, just think about the definition of eigenvalues and eigenvectors, and then imagine what happens in a reflection. --- **Solution Explanation:** To determine the eigenvalues of the matrix \( A \) that represents the reflection across the plane \( 2x + 3y - 5z = 0 \), we consider the geometric properties of reflections and their impact on vectors. 1. **Reflection Plane**: The plane equation given is \( 2x + 3y - 5z = 0 \). Any vector lying on this plane will undergo reflection, typically resulting in it staying in the same plane but possibly changing direction. 2. **Eigenvectors and Eigenvalues**: - An eigenvector is a vector that only scales when a linear transformation is applied, without changing its direction. - The eigenvalue associated with this eigenvector indicates the scale factor. 3. **Considering the Reflection**: - **Perpendicular Vector**: A vector perpendicular to the plane \( 2x + 3y - 5z = 0 \) will change its direction (reflect to the opposite side) when the reflection transformation \( A \) is applied. This means if \( \mathbf{v} \) is perpendicular to the plane, \( A\mathbf{v} = -\mathbf{v} \). Hence, the eigenvalue in this case is \( -1 \). - **Plane Vectors**: Any vector that lies within the plane \( 2x + 3y - 5z = 0 \) remains in the plane after reflection. This implies that these vectors are not changed in direction, though they may get scaled. Essentially, they act as eigenvectors with an eigenvalue of \( 1 \). 4. **Conclusion**: Thus, we have two sets of eigenvalues: - \( 1 \): For eigenvectors lying in the plane of reflection. - \( -1 \): For the eigenvector perpendicular to the plane of reflection. By understanding the geometric implications of
Transcribed Image Text:### Understanding Eigenvalues in the Context of a Reflection Matrix **Question 2:** Suppose that \( A \) is a \( 3 \times 3 \) matrix for the reflection across the plane \( 2x + 3y - 5z = 0 \). What are the eigenvalues of \( A \)? Explain your answer. This problem can be done without any computations, just think about the definition of eigenvalues and eigenvectors, and then imagine what happens in a reflection. --- **Solution Explanation:** To determine the eigenvalues of the matrix \( A \) that represents the reflection across the plane \( 2x + 3y - 5z = 0 \), we consider the geometric properties of reflections and their impact on vectors. 1. **Reflection Plane**: The plane equation given is \( 2x + 3y - 5z = 0 \). Any vector lying on this plane will undergo reflection, typically resulting in it staying in the same plane but possibly changing direction. 2. **Eigenvectors and Eigenvalues**: - An eigenvector is a vector that only scales when a linear transformation is applied, without changing its direction. - The eigenvalue associated with this eigenvector indicates the scale factor. 3. **Considering the Reflection**: - **Perpendicular Vector**: A vector perpendicular to the plane \( 2x + 3y - 5z = 0 \) will change its direction (reflect to the opposite side) when the reflection transformation \( A \) is applied. This means if \( \mathbf{v} \) is perpendicular to the plane, \( A\mathbf{v} = -\mathbf{v} \). Hence, the eigenvalue in this case is \( -1 \). - **Plane Vectors**: Any vector that lies within the plane \( 2x + 3y - 5z = 0 \) remains in the plane after reflection. This implies that these vectors are not changed in direction, though they may get scaled. Essentially, they act as eigenvectors with an eigenvalue of \( 1 \). 4. **Conclusion**: Thus, we have two sets of eigenvalues: - \( 1 \): For eigenvectors lying in the plane of reflection. - \( -1 \): For the eigenvector perpendicular to the plane of reflection. By understanding the geometric implications of
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