Eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}2 & 0 & 1\\0 & 3 & 0\\1 & 0 & 2\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}2 & 0 & 1\\0 & 3 & 0\\1 & 0 & 2\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{ccc}2 - \lambda & 0 & 1\\0 & 3 - \lambda & 0\\1 & 0 & 2 - \lambda\end{array}\right]$$$.

The determinant of the obtained matrix is $$$- \left(\lambda - 3\right)^{2} \left(\lambda - 1\right)$$$ (for steps, see determinant calculator).

Solve the equation $$$- \left(\lambda - 3\right)^{2} \left(\lambda - 1\right) = 0$$$.

The roots are $$$\lambda_{1} = 1$$$, $$$\lambda_{2} = 3$$$, $$$\lambda_{3} = 3$$$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = 1$$$

  $$$\left[\begin{array}{ccc}2 - \lambda & 0 & 1\\0 & 3 - \lambda & 0\\1 & 0 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}1 & 0 & 1\\0 & 2 & 0\\1 & 0 & 1\end{array}\right]$$$

  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.

• $$$\lambda = 3$$$

  $$$\left[\begin{array}{ccc}2 - \lambda & 0 & 1\\0 & 3 - \lambda & 0\\1 & 0 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}-1 & 0 & 1\\0 & 0 & 0\\1 & 0 & -1\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}0\\1\\0\end{array}\right], \left[\begin{array}{c}1\\0\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  These are the eigenvectors.

Eigenvalue: $$$1$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}-1\\0\\1\end{array}\right]$$$A.

Eigenvalue: $$$3$$$A, multiplicity: $$$2$$$A, eigenvectors: $$$\left[\begin{array}{c}0\\1\\0\end{array}\right]$$$, $$$\left[\begin{array}{c}1\\0\\1\end{array}\right]$$$A.