orthogonal matrix Q of eigen vectors
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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![### Problem 5
Consider the real symmetric matrix
\[ A = \begin{pmatrix} 2 & 2 \\ 2 & -1 \end{pmatrix}. \]
**Tasks:**
1. Construct an orthogonal matrix \( Q \) of eigenvectors that would implement a diagonalization of the matrix \( A \).
2. Classify the conic section given by the equation:
\[ 2x_1^2 + 4x_1x_2 - x_2^2 = 6 \]
as an ellipse, parabola, or a hyperbola.
3. Determine the equations of the principal axes of the conic section.
4. Sketch the graph of the conic section.
**Instructions:**
- To find \( Q \), compute the eigenvectors of \( A \) and normalize them to form an orthogonal set.
- Use the eigenvectors to classify the conic section by examining the discriminant of the conic equation.
- Identify any transformations required to align the coordinate axes with the principal axes of the conic section.
- Sketch the graph based on the classification and transformation results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e9683aa-2143-4756-b701-dd7fd0ed0552%2Fead254cd-d2b9-402c-aea0-d5d591667c00%2F8g01op2_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 5
Consider the real symmetric matrix
\[ A = \begin{pmatrix} 2 & 2 \\ 2 & -1 \end{pmatrix}. \]
**Tasks:**
1. Construct an orthogonal matrix \( Q \) of eigenvectors that would implement a diagonalization of the matrix \( A \).
2. Classify the conic section given by the equation:
\[ 2x_1^2 + 4x_1x_2 - x_2^2 = 6 \]
as an ellipse, parabola, or a hyperbola.
3. Determine the equations of the principal axes of the conic section.
4. Sketch the graph of the conic section.
**Instructions:**
- To find \( Q \), compute the eigenvectors of \( A \) and normalize them to form an orthogonal set.
- Use the eigenvectors to classify the conic section by examining the discriminant of the conic equation.
- Identify any transformations required to align the coordinate axes with the principal axes of the conic section.
- Sketch the graph based on the classification and transformation results.
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