Eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}5 & 8 & 16\\4 & 1 & 8\\-4 & -4 & -11\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}5 & 8 & 16\\4 & 1 & 8\\-4 & -4 & -11\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{ccc}5 - \lambda & 8 & 16\\4 & 1 - \lambda & 8\\-4 & -4 & - \lambda - 11\end{array}\right]$$$.

The determinant of the obtained matrix is $$$- \left(\lambda - 1\right) \left(\lambda + 3\right)^{2}$$$ (for steps, see determinant calculator).

Solve the equation $$$- \left(\lambda - 1\right) \left(\lambda + 3\right)^{2} = 0$$$.

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

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = 1$$$

  $$$\left[\begin{array}{ccc}5 - \lambda & 8 & 16\\4 & 1 - \lambda & 8\\-4 & -4 & - \lambda - 11\end{array}\right] = \left[\begin{array}{ccc}4 & 8 & 16\\4 & 0 & 8\\-4 & -4 & -12\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}-2\\-1\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

• $$$\lambda = -3$$$

  $$$\left[\begin{array}{ccc}5 - \lambda & 8 & 16\\4 & 1 - \lambda & 8\\-4 & -4 & - \lambda - 11\end{array}\right] = \left[\begin{array}{ccc}8 & 8 & 16\\4 & 4 & 8\\-4 & -4 & -8\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}-1\\1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  These are the eigenvectors.

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

Eigenvalue: $$$-3$$$A, multiplicity: $$$2$$$A, eigenvectors: $$$\left[\begin{array}{c}-1\\1\\0\end{array}\right]$$$, $$$\left[\begin{array}{c}-2\\0\\1\end{array}\right]$$$A.