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]
