Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To find the eigenvalues and the eigenvector corresponding to the largest eigenvalue for the given matrix \( A \), follow these steps:
1. Define the matrix \( A \):
[tex]\[ A = \begin{pmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{pmatrix} \][/tex]
2. Compute the characteristic polynomial:
The characteristic polynomial is obtained by solving \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix and \(\lambda\) represents the eigenvalues.
3. Solve for the eigenvalues:
By solving the characteristic polynomial equation, we find the eigenvalues. The eigenvalues for this matrix are:
[tex]\[ \lambda_1 = -2, \quad \lambda_2 = 3, \quad \lambda_3 = 6 \][/tex]
4. Identify the largest eigenvalue:
Among the eigenvalues \(-2\), \(3\), and \(6\), the largest eigenvalue is \(6\).
5. Find the eigenvector corresponding to the largest eigenvalue:
To find the eigenvector corresponding to the largest eigenvalue (\(\lambda = 6\)), substitute \(\lambda = 6\) into the equation:
[tex]\[ (A - 6I) \mathbf{v} = 0 \][/tex]
This simplifies into solving the system of linear equations represented by:
[tex]\[ \begin{pmatrix} 1 - 6 & 1 & 3 \\ 1 & 5 - 6 & 1 \\ 3 & 1 & 1 - 6 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = 0 \][/tex]
Simplifying the matrix:
[tex]\[ \begin{pmatrix} -5 & 1 & 3 \\ 1 & -1 & 1 \\ 3 & 1 & -5 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = 0 \][/tex]
6. Solve for the eigenvector components:
Solving this system leads us to find the eigenvector corresponding to \(\lambda = 6\). The eigenvector for the largest eigenvalue \(\lambda = 6\) is:
[tex]\[ \mathbf{v} = \begin{pmatrix} -0.40824829 \\ -0.81649658 \\ -0.40824829 \end{pmatrix} \][/tex]
So, to summarize:
- The eigenvalues of the matrix \(A\) are:
[tex]\[ \{ -2, 3, 6 \} \][/tex]
- The largest eigenvalue is:
[tex]\[ 6 \][/tex]
- The eigenvector corresponding to the largest eigenvalue (\(\lambda = 6\)) is:
[tex]\[ \begin{pmatrix} -0.40824829 \\ -0.81649658 \\ -0.40824829 \end{pmatrix} \][/tex]
1. Define the matrix \( A \):
[tex]\[ A = \begin{pmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{pmatrix} \][/tex]
2. Compute the characteristic polynomial:
The characteristic polynomial is obtained by solving \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix and \(\lambda\) represents the eigenvalues.
3. Solve for the eigenvalues:
By solving the characteristic polynomial equation, we find the eigenvalues. The eigenvalues for this matrix are:
[tex]\[ \lambda_1 = -2, \quad \lambda_2 = 3, \quad \lambda_3 = 6 \][/tex]
4. Identify the largest eigenvalue:
Among the eigenvalues \(-2\), \(3\), and \(6\), the largest eigenvalue is \(6\).
5. Find the eigenvector corresponding to the largest eigenvalue:
To find the eigenvector corresponding to the largest eigenvalue (\(\lambda = 6\)), substitute \(\lambda = 6\) into the equation:
[tex]\[ (A - 6I) \mathbf{v} = 0 \][/tex]
This simplifies into solving the system of linear equations represented by:
[tex]\[ \begin{pmatrix} 1 - 6 & 1 & 3 \\ 1 & 5 - 6 & 1 \\ 3 & 1 & 1 - 6 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = 0 \][/tex]
Simplifying the matrix:
[tex]\[ \begin{pmatrix} -5 & 1 & 3 \\ 1 & -1 & 1 \\ 3 & 1 & -5 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = 0 \][/tex]
6. Solve for the eigenvector components:
Solving this system leads us to find the eigenvector corresponding to \(\lambda = 6\). The eigenvector for the largest eigenvalue \(\lambda = 6\) is:
[tex]\[ \mathbf{v} = \begin{pmatrix} -0.40824829 \\ -0.81649658 \\ -0.40824829 \end{pmatrix} \][/tex]
So, to summarize:
- The eigenvalues of the matrix \(A\) are:
[tex]\[ \{ -2, 3, 6 \} \][/tex]
- The largest eigenvalue is:
[tex]\[ 6 \][/tex]
- The eigenvector corresponding to the largest eigenvalue (\(\lambda = 6\)) is:
[tex]\[ \begin{pmatrix} -0.40824829 \\ -0.81649658 \\ -0.40824829 \end{pmatrix} \][/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
