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

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

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

The determinant of the obtained matrix is $$$\lambda^{2} - 3 \lambda t^{2} + t^{4}$$$ (for steps, see determinant calculator).

Solve the equation $$$\lambda^{2} - 3 \lambda t^{2} + t^{4} = 0$$$.

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

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = \frac{t^{2} \left(3 - \sqrt{5}\right)}{2}$$$

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

  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.

• $$$\lambda = \frac{t^{2} \left(\sqrt{5} + 3\right)}{2}$$$

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

  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: $$$\frac{t^{2} \left(3 - \sqrt{5}\right)}{2}\approx 0.381966011250105 t^{2}$$$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: $$$\frac{t^{2} \left(\sqrt{5} + 3\right)}{2}\approx 2.618033988749895 t^{2}$$$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.