**Problem 1: Analysis of a Real Symmetric Matrix**Consider the real symmetric matrix:\[A = \begin{pmatrix} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{pmatrix}.\]Tasks:1. **Determine Eigenvalues and Eigenvectors:** - Calculate the eigenvalues and corresponding eigenvectors of matrix $ A $.2. **Orthogonal Matrix Construction:** - Use the Gram-Schmidt process to construct an orthogonal matrix $ Q $ from the eigenvectors. This orthogonal matrix will facilitate the diagonalization of matrix $ A $.3. **Classification of the Quadratic Form:** - Analyze the quadratic form given by: \[ 3x_1^2 + 6x_2^2 + 3x_3^2 - 4x_1x_2 + 8x_1x_3 + 4x_2x_3 = 98 \] - Identify whether the quadratic form represents: - An ellipsoid - A hyperboloid (one sheet or two sheets) - A paraboloid**Note:** The classification requires examining the coefficients and determining the nature of the conic section represented by the quadratic form.

Given real symmetric matrix is A=3-24-262423 let λ be the Eigenvalue of the given matrix. So…

Consider the real symmetric matrix 3 -2 4 A -2 6. 4 2 3 Determine eigen values and eigen vectors. Use Gram-Schimidt to construct an or- thogonal matrix Q of eigen vectors that would implement a diagonalization. Iden- tify whether

Consider the real symmetric matrix 3 -2 4 A -2 6. 4 2 3 Determine eigen values and eigen vectors. Use Gram-Schimidt to construct an or- thogonal matrix Q of eigen vectors that would implement a diagonalization. Iden- tify whether

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 1: Analysis of a Real Symmetric Matrix**

Consider the real symmetric matrix:

\[
A = \begin{pmatrix}
3 & -2 & 4 \\
-2 & 6 & 2 \\
4 & 2 & 3
\end{pmatrix}.
\]

Tasks:

1. **Determine Eigenvalues and Eigenvectors:**
- Calculate the eigenvalues and corresponding eigenvectors of matrix $ A $.

2. **Orthogonal Matrix Construction:**
- Use the Gram-Schmidt process to construct an orthogonal matrix $ Q $ from the eigenvectors. This orthogonal matrix will facilitate the diagonalization of matrix $ A $.

3. **Classification of the Quadratic Form:**
- Analyze the quadratic form given by:
\[
3x_1^2 + 6x_2^2 + 3x_3^2 - 4x_1x_2 + 8x_1x_3 + 4x_2x_3 = 98
\]
- Identify whether the quadratic form represents:
- An ellipsoid
- A hyperboloid (one sheet or two sheets)
- A paraboloid

**Note:** The classification requires examining the coefficients and determining the nature of the conic section represented by the quadratic form.

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