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

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

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

The determinant of the obtained matrix is $$$\left(\lambda - 4\right)^{2} \left(\lambda^{2} - 8 \lambda + 10\right)$$$ (for steps, see determinant calculator).

Solve the equation $$$\left(\lambda - 4\right)^{2} \left(\lambda^{2} - 8 \lambda + 10\right) = 0$$$.

The roots are $$$\lambda_{1} = 4$$$, $$$\lambda_{2} = 4$$$, $$$\lambda_{3} = 4 - \sqrt{6}$$$, $$$\lambda_{4} = \sqrt{6} + 4$$$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = 4$$$

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

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

  These are the eigenvectors.

• $$$\lambda = 4 - \sqrt{6}$$$

  $$$\left[\begin{array}{cccc}4 - \lambda & 0 & 1 & 0\\0 & 4 - \lambda & 1 & 0\\1 & 1 & 4 - \lambda & 2\\0 & 0 & 2 & 4 - \lambda\end{array}\right] = \left[\begin{array}{cccc}\sqrt{6} & 0 & 1 & 0\\0 & \sqrt{6} & 1 & 0\\1 & 1 & \sqrt{6} & 2\\0 & 0 & 2 & \sqrt{6}\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}\frac{1}{2}\\\frac{1}{2}\\- \frac{\sqrt{6}}{2}\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

• $$$\lambda = \sqrt{6} + 4$$$

  $$$\left[\begin{array}{cccc}4 - \lambda & 0 & 1 & 0\\0 & 4 - \lambda & 1 & 0\\1 & 1 & 4 - \lambda & 2\\0 & 0 & 2 & 4 - \lambda\end{array}\right] = \left[\begin{array}{cccc}- \sqrt{6} & 0 & 1 & 0\\0 & - \sqrt{6} & 1 & 0\\1 & 1 & - \sqrt{6} & 2\\0 & 0 & 2 & - \sqrt{6}\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}\frac{1}{2}\\\frac{1}{2}\\\frac{\sqrt{6}}{2}\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

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

Eigenvalue: $$$4 - \sqrt{6}\approx 1.550510257216822$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}\frac{1}{2}\\\frac{1}{2}\\- \frac{\sqrt{6}}{2}\\1\end{array}\right]\approx \left[\begin{array}{c}0.5\\0.5\\-1.224744871391589\\1\end{array}\right]$$$A.

Eigenvalue: $$$\sqrt{6} + 4\approx 6.449489742783178$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}\frac{1}{2}\\\frac{1}{2}\\\frac{\sqrt{6}}{2}\\1\end{array}\right]\approx \left[\begin{array}{c}0.5\\0.5\\1.224744871391589\\1\end{array}\right]$$$A.