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Find the eigenvalues and the eigenvector for the largest eigenvalue of the matrix [tex]A=\left[\begin{array}{lll}1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1\end{array}\right][/tex].

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]