Eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}1 & 1 & 3\\1 & 5 & 1\\3 & 1 & 1\end{array}\right]$$$
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Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}1 & 1 & 3\\1 & 5 & 1\\3 & 1 & 1\end{array}\right]$$$.
Solution
Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{ccc}1 - \lambda & 1 & 3\\1 & 5 - \lambda & 1\\3 & 1 & 1 - \lambda\end{array}\right]$$$.
The determinant of the obtained matrix is $$$- \lambda^{3} + 7 \lambda^{2} - 36$$$ (for steps, see determinant calculator).
Solve the equation $$$- \lambda^{3} + 7 \lambda^{2} - 36 = 0$$$.
The roots are $$$\lambda_{1} = 6$$$, $$$\lambda_{2} = 3$$$, $$$\lambda_{3} = -2$$$ (for steps, see equation solver).
These are the eigenvalues.
Next, find the eigenvectors.
$$$\lambda = 6$$$
$$$\left[\begin{array}{ccc}1 - \lambda & 1 & 3\\1 & 5 - \lambda & 1\\3 & 1 & 1 - \lambda\end{array}\right] = \left[\begin{array}{ccc}-5 & 1 & 3\\1 & -1 & 1\\3 & 1 & -5\end{array}\right]$$$
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.
$$$\lambda = 3$$$
$$$\left[\begin{array}{ccc}1 - \lambda & 1 & 3\\1 & 5 - \lambda & 1\\3 & 1 & 1 - \lambda\end{array}\right] = \left[\begin{array}{ccc}-2 & 1 & 3\\1 & 2 & 1\\3 & 1 & -2\end{array}\right]$$$
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.
$$$\lambda = -2$$$
$$$\left[\begin{array}{ccc}1 - \lambda & 1 & 3\\1 & 5 - \lambda & 1\\3 & 1 & 1 - \lambda\end{array}\right] = \left[\begin{array}{ccc}3 & 1 & 3\\1 & 7 & 1\\3 & 1 & 3\end{array}\right]$$$
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: $$$6$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}1\\2\\1\end{array}\right]$$$A.
Eigenvalue: $$$3$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}1\\-1\\1\end{array}\right]$$$A.
Eigenvalue: $$$-2$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}-1\\0\\1\end{array}\right]$$$A.