Eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}2 & 1 & 4\\0 & 2 & 0\\1 & 1 & 2\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}2 & 1 & 4\\0 & 2 & 0\\1 & 1 & 2\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{ccc}2 - \lambda & 1 & 4\\0 & 2 - \lambda & 0\\1 & 1 & 2 - \lambda\end{array}\right]$$$.

The determinant of the obtained matrix is $$$- \lambda \left(\lambda - 4\right) \left(\lambda - 2\right)$$$ (for steps, see determinant calculator).

Solve the equation $$$- \lambda \left(\lambda - 4\right) \left(\lambda - 2\right) = 0$$$.

The roots are $$$\lambda_{1} = 4$$$, $$$\lambda_{2} = 2$$$, $$$\lambda_{3} = 0$$$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = 4$$$

  $$$\left[\begin{array}{ccc}2 - \lambda & 1 & 4\\0 & 2 - \lambda & 0\\1 & 1 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}-2 & 1 & 4\\0 & -2 & 0\\1 & 1 & -2\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}2\\0\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

• $$$\lambda = 2$$$

  $$$\left[\begin{array}{ccc}2 - \lambda & 1 & 4\\0 & 2 - \lambda & 0\\1 & 1 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}0 & 1 & 4\\0 & 0 & 0\\1 & 1 & 0\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}4\\-4\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

• $$$\lambda = 0$$$

  $$$\left[\begin{array}{ccc}2 - \lambda & 1 & 4\\0 & 2 - \lambda & 0\\1 & 1 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}2 & 1 & 4\\0 & 2 & 0\\1 & 1 & 2\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

Eigenvalue: $$$4$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}2\\0\\1\end{array}\right]$$$A.

Eigenvalue: $$$2$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}4\\-4\\1\end{array}\right]$$$A.

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