Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest.) 0 3 3 30 3 3 3 A₁ = For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.) dim(x) =
Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest.) 0 3 3 30 3 3 3 A₁ = For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.) dim(x) =
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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![### Eigenvalues and Eigenspaces of a Symmetric Matrix
#### Problem Statement
**Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest.)**
\[
\begin{pmatrix}
0 & 3 & 3 \\
3 & 0 & 3 \\
3 & 3 & 3
\end{pmatrix}
\]
\[
\lambda_i = \boxed{}
\]
**For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.)**
\[
\text{dim}(E_{\lambda_i}) = \boxed{}
\]
#### Explanation:
In this problem, you are provided with a symmetric 3x3 matrix and are asked to find its eigenvalues in ascending order.
- **Eigenvalues (\(\lambda_i\))**: These are special numbers associated with the matrix that indicate the scaler by which the eigenvector is stretched under the matrix transformation. For a symmetric matrix, all eigenvalues are real.
- **Eigenspace (\(E_{\lambda_i}\))**: For each eigenvalue \(\lambda_i\), the eigenspace is the set of all eigenvectors corresponding to that eigenvalue, along with the zero vector. The dimension of the eigenspace indicates the number of linearly independent eigenvectors corresponding to \(\lambda_i\).
The steps to solve this are:
1. **Find the Eigenvalues**: Calculate the characteristic polynomial of the matrix and solve it to find the eigenvalues.
2. **Find the Dimensions of the Eigenspaces**: Determine the number of linearly independent eigenvectors corresponding to each eigenvalue, i.e., the dimension of each eigenspace.
By performing these steps, you will be able to find the eigenvalues and their corresponding eigenspaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03210149-7bd1-493b-9cd1-96815f185546%2Ffdcdd1bc-90a1-4cdc-b756-f902925305c2%2Ftsapxzq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Eigenvalues and Eigenspaces of a Symmetric Matrix
#### Problem Statement
**Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest.)**
\[
\begin{pmatrix}
0 & 3 & 3 \\
3 & 0 & 3 \\
3 & 3 & 3
\end{pmatrix}
\]
\[
\lambda_i = \boxed{}
\]
**For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.)**
\[
\text{dim}(E_{\lambda_i}) = \boxed{}
\]
#### Explanation:
In this problem, you are provided with a symmetric 3x3 matrix and are asked to find its eigenvalues in ascending order.
- **Eigenvalues (\(\lambda_i\))**: These are special numbers associated with the matrix that indicate the scaler by which the eigenvector is stretched under the matrix transformation. For a symmetric matrix, all eigenvalues are real.
- **Eigenspace (\(E_{\lambda_i}\))**: For each eigenvalue \(\lambda_i\), the eigenspace is the set of all eigenvectors corresponding to that eigenvalue, along with the zero vector. The dimension of the eigenspace indicates the number of linearly independent eigenvectors corresponding to \(\lambda_i\).
The steps to solve this are:
1. **Find the Eigenvalues**: Calculate the characteristic polynomial of the matrix and solve it to find the eigenvalues.
2. **Find the Dimensions of the Eigenspaces**: Determine the number of linearly independent eigenvectors corresponding to each eigenvalue, i.e., the dimension of each eigenspace.
By performing these steps, you will be able to find the eigenvalues and their corresponding eigenspaces.
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