Eigenvalues and eigenvectors of $\left[\begin{array}{cc}8 & 8\\8 & 8\end{array}\right]$

Find the eigenvalues and eigenvectors of $\left[\begin{array}{cc}8 & 8\\8 & 8\end{array}\right]$.

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

The determinant of the obtained matrix is $\lambda \left(\lambda - 16\right)$ (for steps, see determinant calculator).

Solve the equation $\lambda \left(\lambda - 16\right) = 0$.

The roots are $\lambda_{1} = 16$, $\lambda_{2} = 0$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

• $\lambda = 16$

  $\left[\begin{array}{cc}8 - \lambda & 8\\8 & 8 - \lambda\end{array}\right] = \left[\begin{array}{cc}-8 & 8\\8 & -8\end{array}\right]$

  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.

• $\lambda = 0$

  $\left[\begin{array}{cc}8 - \lambda & 8\\8 & 8 - \lambda\end{array}\right] = \left[\begin{array}{cc}8 & 8\\8 & 8\end{array}\right]$

  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: $16$A, multiplicity: $1$A, eigenvector: $\left[\begin{array}{c}1\\1\end{array}\right]$A.

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