Eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}\frac{2}{3} & 0\\0 & \frac{1}{3}\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}\frac{2}{3} & 0\\0 & \frac{1}{3}\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{cc}\frac{2}{3} - \lambda & 0\\0 & \frac{1}{3} - \lambda\end{array}\right]$$$.

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

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

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

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = \frac{2}{3}$$$

  $$$\left[\begin{array}{cc}\frac{2}{3} - \lambda & 0\\0 & \frac{1}{3} - \lambda\end{array}\right] = \left[\begin{array}{cc}0 & 0\\0 & - \frac{1}{3}\end{array}\right]$$$

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

  This is the eigenvector.

• $$$\lambda = \frac{1}{3}$$$

  $$$\left[\begin{array}{cc}\frac{2}{3} - \lambda & 0\\0 & \frac{1}{3} - \lambda\end{array}\right] = \left[\begin{array}{cc}\frac{1}{3} & 0\\0 & 0\end{array}\right]$$$

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

  This is the eigenvector.

Eigenvalue: $$$\frac{2}{3}\approx 0.666666666666667$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}1\\0\end{array}\right]$$$A.

Eigenvalue: $$$\frac{1}{3}\approx 0.333333333333333$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}0\\1\end{array}\right]$$$A.