The image contains mathematical content involving linear algebra concepts such as matrices, eigenvalues, eigenvectors, orthogonal diagonalization, and spectral decomposition. Below is the transcription tailored for an educational website:---### Linear Algebra: Matrices, Eigenvalues, and EigenvectorsGiven the matrix $ A $:\[ A = \begin{bmatrix}3 & -2 & 4 \\-2 & 6 & 2 \\4 & 2 & 3\end{bmatrix} \]The eigenvalues and corresponding eigenspaces are calculated as follows:For the eigenvalue $ \lambda = 7 $:\[ \lambda = 7: \; \begin{Bmatrix}\begin{pmatrix}1 \\0 \\1\end{pmatrix}, \begin{pmatrix}-1/2 \\1 \\0\end{pmatrix}\end{Bmatrix} \]For the eigenvalue $ \lambda = -2 $:\[ \lambda = -2: \; \begin{Bmatrix}\begin{pmatrix}-1 \\-1/2 \\1\end{pmatrix}\end{Bmatrix} \]### Exercises**(a)** Orthogonally diagonalize the matrix $ A $.**(b)** Construct a spectral decomposition of the matrix $ A $.This matrix analysis involves understanding how to calculate eigenvalues and eigenvectors, and then using these to transform the matrix into a diagonal form, as well as decomposing it spectrally.---In the explanation, we have highlighted both eigenvalues and their corresponding eigenspaces, which help in further operations such as orthogonal diagonalization and spectral decomposition. This setup provides a clear understanding of the matrix $ A $ and its properties for further analysis in linear algebra studies.

Answered: 2. The eigenvalues 31 Let A =-2 6 4 2…

2. The eigenvalues 31 Let A =-2 6 4 2 and corresponding eigenspaces are: -1 2 = 7: 1 and 1 = -2: 1/2 (a) Orthogonally diagonalize the matrix A. (b) Construct a spectral decomposition of the matrix A.

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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The image contains mathematical content involving linear algebra concepts such as matrices, eigenvalues, eigenvectors, orthogonal diagonalization, and spectral decomposition. Below is the transcription tailored for an educational website:

---

### Linear Algebra: Matrices, Eigenvalues, and Eigenvectors

Given the matrix $ A $:

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

The eigenvalues and corresponding eigenspaces are calculated as follows:

For the eigenvalue $ \lambda = 7 $:

\[ \lambda = 7: \; \begin{Bmatrix}
\begin{pmatrix}
1 \\
0 \\
1
\end{pmatrix},
\begin{pmatrix}
-1/2 \\
1 \\
0
\end{pmatrix}
\end{Bmatrix} \]

For the eigenvalue $ \lambda = -2 $:

\[ \lambda = -2: \; \begin{Bmatrix}
\begin{pmatrix}
-1 \\
-1/2 \\
1
\end{pmatrix}
\end{Bmatrix} \]

### Exercises

**(a)** Orthogonally diagonalize the matrix $ A $.

**(b)** Construct a spectral decomposition of the matrix $ A $.

This matrix analysis involves understanding how to calculate eigenvalues and eigenvectors, and then using these to transform the matrix into a diagonal form, as well as decomposing it spectrally.

---

In the explanation, we have highlighted both eigenvalues and their corresponding eigenspaces, which help in further operations such as orthogonal diagonalization and spectral decomposition. This setup provides a clear understanding of the matrix $ A $ and its properties for further analysis in linear algebra studies.

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