Eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}2 & 2 & 2\\2 & 6 & 2\\2 & 2 & 2\end{array}\right]$$$
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Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}2 & 2 & 2\\2 & 6 & 2\\2 & 2 & 2\end{array}\right]$$$.
Solution
Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{ccc}2 - \lambda & 2 & 2\\2 & 6 - \lambda & 2\\2 & 2 & 2 - \lambda\end{array}\right]$$$.
The determinant of the obtained matrix is $$$- \lambda \left(\lambda - 8\right) \left(\lambda - 2\right)$$$ (for steps, see determinant calculator).
Solve the equation $$$- \lambda \left(\lambda - 8\right) \left(\lambda - 2\right) = 0$$$.
The roots are $$$\lambda_{1} = 8$$$, $$$\lambda_{2} = 2$$$, $$$\lambda_{3} = 0$$$ (for steps, see equation solver).
These are the eigenvalues.
Next, find the eigenvectors.
$$$\lambda = 8$$$
$$$\left[\begin{array}{ccc}2 - \lambda & 2 & 2\\2 & 6 - \lambda & 2\\2 & 2 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}-6 & 2 & 2\\2 & -2 & 2\\2 & 2 & -6\end{array}\right]$$$
The null space of this matrix is $$$\left\{\left[\begin{array}{c}1\\2\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).
This is the eigenvector.
$$$\lambda = 2$$$
$$$\left[\begin{array}{ccc}2 - \lambda & 2 & 2\\2 & 6 - \lambda & 2\\2 & 2 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}0 & 2 & 2\\2 & 4 & 2\\2 & 2 & 0\end{array}\right]$$$
The null space of this matrix is $$$\left\{\left[\begin{array}{c}1\\-1\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).
This is the eigenvector.
$$$\lambda = 0$$$
$$$\left[\begin{array}{ccc}2 - \lambda & 2 & 2\\2 & 6 - \lambda & 2\\2 & 2 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}2 & 2 & 2\\2 & 6 & 2\\2 & 2 & 2\end{array}\right]$$$
The null space of this matrix is $$$\left\{\left[\begin{array}{c}-1\\0\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).
This is the eigenvector.
Answer
Eigenvalue: $$$8$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}1\\2\\1\end{array}\right]$$$A.
Eigenvalue: $$$2$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}1\\-1\\1\end{array}\right]$$$A.
Eigenvalue: $$$0$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}-1\\0\\1\end{array}\right]$$$A.