Eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}\frac{5}{2} & \frac{3}{2}\\- \frac{3}{2} & - \frac{1}{2}\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}\frac{5}{2} & \frac{3}{2}\\- \frac{3}{2} & - \frac{1}{2}\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{cc}\frac{5}{2} - \lambda & \frac{3}{2}\\- \frac{3}{2} & - \lambda - \frac{1}{2}\end{array}\right]$$$.

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

Solve the equation $$$\left(\lambda - 1\right)^{2} = 0$$$.

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

These are the eigenvalues.

Next, find the eigenvectors.

$$$\lambda = 1$$$

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

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