To find the eigenvalues and eigenvectors for the matrix [tex]\( A \)[/tex], we need to perform the following steps:
1. Setup the matrix equation:
The characteristic equation of a matrix [tex]\( A \)[/tex] is given by [tex]\( \det(A - \lambda I) = 0 \)[/tex], where [tex]\( \lambda \)[/tex] represents the eigenvalues and [tex]\( I \)[/tex] is the identity matrix of the same dimension as [tex]\( A \)[/tex].
2. Compute the determinant:
For our 3x3 matrix [tex]\( A \)[/tex], we can set up [tex]\( A - \lambda I \)[/tex] where [tex]\( \lambda I \)[/tex] subtracts [tex]\( \lambda \)[/tex] from the diagonal elements of [tex]\( A \)[/tex]:
[tex]\[
A - \lambda I = \left[\begin{array}{ccc}
2 - \lambda & 1 & 3 \\
1 & 2 - \lambda & 3 \\
3 & 3 & 20 - \lambda
\end{array}\right]
\][/tex]
The characteristic equation is [tex]\( \det(A - \lambda I) = 0 \)[/tex].
3. Solve for [tex]\( \lambda \)[/tex]:
By solving the determinant, we obtain the eigenvalues. The eigenvalues of the given matrix [tex]\( A \)[/tex] are:
[tex]\[
\lambda_1 = 21, \quad \lambda_2 = 1, \quad \lambda_3 = 2
\][/tex]
4. Determine the eigenvectors:
To find the eigenvectors corresponding to each [tex]\( \lambda \)[/tex], we solve the system of linear equations given by [tex]\( (A - \lambda I)\mathbf{v} = 0 \)[/tex] for each eigenvalue [tex]\( \lambda \)[/tex], where [tex]\( \mathbf{v} \)[/tex] is the eigenvector.
For [tex]\( \lambda_1 = 21 \)[/tex]:
[tex]\[
\mathbf{v}_1 = \begin{bmatrix}
-0.16222142 \\
-0.16222142 \\
-0.97332853
\end{bmatrix}
\][/tex]
For [tex]\( \lambda_2 = 1 \)[/tex]:
[tex]\[
\mathbf{v}_2 = \begin{bmatrix}
-0.70710678 \\
0.70710678 \\
-1.67707020 \times 10^{-16}
\end{bmatrix}
\][/tex]
For [tex]\( \lambda_3 = 2 \)[/tex]:
[tex]\[
\mathbf{v}_3 = \begin{bmatrix}
-0.68824720 \\
-0.68824720 \\
0.22941573
\end{bmatrix}
\][/tex]
Therefore, the eigenvalues and their corresponding eigenvectors for matrix [tex]\( A \)[/tex] are:
- Eigenvalue [tex]\( \lambda_1 = 21 \)[/tex] with eigenvector [tex]\(\mathbf{v}_1 = \begin{bmatrix} -0.16222142 \\ -0.16222142 \\ -0.97332853 \end{bmatrix} \)[/tex]
- Eigenvalue [tex]\( \lambda_2 = 1 \)[/tex] with eigenvector [tex]\(\mathbf{v}_2 = \begin{bmatrix} -0.70710678 \\ 0.70710678 \\ -1.67707020 \times 10^{-16} \end{bmatrix} \)[/tex]
- Eigenvalue [tex]\( \lambda_3 = 2 \)[/tex] with eigenvector [tex]\(\mathbf{v}_3 = \begin{bmatrix} -0.68824720 \\ -0.68824720 \\ 0.22941573 \end{bmatrix} \)[/tex]
