Eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}1 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}1 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{ccc}1 - \lambda & 0 & 0\\0 & - \lambda & 1\\0 & 1 & - \lambda\end{array}\right]$$$.

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

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

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

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = -1$$$

  $$$\left[\begin{array}{ccc}1 - \lambda & 0 & 0\\0 & - \lambda & 1\\0 & 1 & - \lambda\end{array}\right] = \left[\begin{array}{ccc}2 & 0 & 0\\0 & 1 & 1\\0 & 1 & 1\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}0\\-1\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

• $$$\lambda = 1$$$

  $$$\left[\begin{array}{ccc}1 - \lambda & 0 & 0\\0 & - \lambda & 1\\0 & 1 & - \lambda\end{array}\right] = \left[\begin{array}{ccc}0 & 0 & 0\\0 & -1 & 1\\0 & 1 & -1\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}1\\0\\0\end{array}\right], \left[\begin{array}{c}0\\1\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  These are the eigenvectors.

Eigenvalue: $$$-1$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}0\\-1\\1\end{array}\right]$$$A.

Eigenvalue: $$$1$$$A, multiplicity: $$$2$$$A, eigenvectors: $$$\left[\begin{array}{c}1\\0\\0\end{array}\right]$$$, $$$\left[\begin{array}{c}0\\1\\1\end{array}\right]$$$A.