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

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

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

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

Solve the equation $$$\lambda^{2} - 3 \lambda + 1 = 0$$$.

The roots are $$$\lambda_{1} = - \frac{-3 + \sqrt{5}}{2}$$$, $$$\lambda_{2} = \frac{\sqrt{5} + 3}{2}$$$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = - \frac{-3 + \sqrt{5}}{2}$$$

  $$$\left[\begin{array}{cc}2 - \lambda & 1\\1 & 1 - \lambda\end{array}\right] = \left[\begin{array}{cc}\frac{-3 + \sqrt{5}}{2} + 2 & 1\\1 & \frac{-3 + \sqrt{5}}{2} + 1\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}- \frac{-1 + \sqrt{5}}{2}\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

• $$$\lambda = \frac{\sqrt{5} + 3}{2}$$$

  $$$\left[\begin{array}{cc}2 - \lambda & 1\\1 & 1 - \lambda\end{array}\right] = \left[\begin{array}{cc}2 - \frac{\sqrt{5} + 3}{2} & 1\\1 & 1 - \frac{\sqrt{5} + 3}{2}\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}\frac{1 + \sqrt{5}}{2}\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

Eigenvalue: $$$- \frac{-3 + \sqrt{5}}{2}\approx 0.381966011250105$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}- \frac{-1 + \sqrt{5}}{2}\\1\end{array}\right]\approx \left[\begin{array}{c}-0.618033988749895\\1\end{array}\right]$$$A.

Eigenvalue: $$$\frac{\sqrt{5} + 3}{2}\approx 2.618033988749895$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}\frac{1 + \sqrt{5}}{2}\\1\end{array}\right]\approx \left[\begin{array}{c}1.618033988749895\\1\end{array}\right]$$$A.