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.
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
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15c5ec76-8bab-480b-9121-367c8408675d%2F6f34d6f6-9d83-4719-86ae-0a2b0a2ed9c4%2Fjll0jm_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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