Eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}2 & 2 & 2\\2 & 6 & 2\\2 & 2 & 2\end{array}\right]$$$

The calculator will find the eigenvalues and eigenvectors of the square $$$3$$$x$$$3$$$ matrix $$$\left[\begin{array}{ccc}2 & 2 & 2\\2 & 6 & 2\\2 & 2 & 2\end{array}\right]$$$, with steps shown.

Related calculator: Characteristic Polynomial Calculator

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your Input

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{ccc}2 & 2 & 2\\2 & 6 & 2\\2 & 2 & 2\end{array}\right]$$$.

Solution

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{ccc}2 - \lambda & 2 & 2\\2 & 6 - \lambda & 2\\2 & 2 & 2 - \lambda\end{array}\right]$$$.

The determinant of the obtained matrix is $$$- \lambda \left(\lambda - 8\right) \left(\lambda - 2\right)$$$ (for steps, see determinant calculator).

Solve the equation $$$- \lambda \left(\lambda - 8\right) \left(\lambda - 2\right) = 0$$$.

The roots are $$$\lambda_{1} = 8$$$, $$$\lambda_{2} = 2$$$, $$$\lambda_{3} = 0$$$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

  • $$$\lambda = 8$$$

    $$$\left[\begin{array}{ccc}2 - \lambda & 2 & 2\\2 & 6 - \lambda & 2\\2 & 2 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}-6 & 2 & 2\\2 & -2 & 2\\2 & 2 & -6\end{array}\right]$$$

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

    This is the eigenvector.

  • $$$\lambda = 2$$$

    $$$\left[\begin{array}{ccc}2 - \lambda & 2 & 2\\2 & 6 - \lambda & 2\\2 & 2 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}0 & 2 & 2\\2 & 4 & 2\\2 & 2 & 0\end{array}\right]$$$

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

    This is the eigenvector.

  • $$$\lambda = 0$$$

    $$$\left[\begin{array}{ccc}2 - \lambda & 2 & 2\\2 & 6 - \lambda & 2\\2 & 2 & 2 - \lambda\end{array}\right] = \left[\begin{array}{ccc}2 & 2 & 2\\2 & 6 & 2\\2 & 2 & 2\end{array}\right]$$$

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

    This is the eigenvector.

Answer

Eigenvalue: $$$8$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}1\\2\\1\end{array}\right]$$$A.

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

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