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

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

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

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

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

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

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = -2$$$

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

  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.

• $$$\lambda = 2$$$

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

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}3\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

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

Eigenvalue: $$$2$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}3\\1\end{array}\right]$$$A.