### Problem 5Consider 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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Answered: orthogonal matrix Q of eigen vectors

Math

Advanced Math

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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Question

### 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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