Eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}1 & \frac{1}{5}\\\frac{1}{5} & 1\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}1 & \frac{1}{5}\\\frac{1}{5} & 1\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{cc}1 - \lambda & \frac{1}{5}\\\frac{1}{5} & 1 - \lambda\end{array}\right]$$$.

The determinant of the obtained matrix is $$$\lambda^{2} - 2 \lambda + \frac{24}{25}$$$ (for steps, see determinant calculator).

Solve the equation $$$\lambda^{2} - 2 \lambda + \frac{24}{25} = 0$$$.

The roots are $$$\lambda_{1} = \frac{6}{5}$$$, $$$\lambda_{2} = \frac{4}{5}$$$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = \frac{6}{5}$$$

  $$$\left[\begin{array}{cc}1 - \lambda & \frac{1}{5}\\\frac{1}{5} & 1 - \lambda\end{array}\right] = \left[\begin{array}{cc}- \frac{1}{5} & \frac{1}{5}\\\frac{1}{5} & - \frac{1}{5}\end{array}\right]$$$

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

  This is the eigenvector.

• $$$\lambda = \frac{4}{5}$$$

  $$$\left[\begin{array}{cc}1 - \lambda & \frac{1}{5}\\\frac{1}{5} & 1 - \lambda\end{array}\right] = \left[\begin{array}{cc}\frac{1}{5} & \frac{1}{5}\\\frac{1}{5} & \frac{1}{5}\end{array}\right]$$$

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

  This is the eigenvector.

Eigenvalue: $$$\frac{6}{5} = 1.2$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}1\\1\end{array}\right]$$$A.

Eigenvalue: $$$\frac{4}{5} = 0.8$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}-1\\1\end{array}\right]$$$A.