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.