Eigenvalues and Eigenvectors Calculator
The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown.
Related calculator: Characteristic Polynomial Calculator
Your Input
Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}1 & 2\\0 & 3\end{array}\right]$$$.
Solution
Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{cc}1 - \lambda & 2\\0 & 3 - \lambda\end{array}\right]$$$.
The determinant of the obtained matrix is $$$\left(\lambda - 3\right) \left(\lambda - 1\right)$$$ (for steps, see determinant calculator).
Solve the equation $$$\left(\lambda - 3\right) \left(\lambda - 1\right) = 0$$$.
The roots are $$$\lambda_{1} = 3$$$, $$$\lambda_{2} = 1$$$ (for steps, see equation solver).
These are the eigenvalues.
Next, find the eigenvectors.
$$$\lambda = 3$$$
$$$\left[\begin{array}{cc}1 - \lambda & 2\\0 & 3 - \lambda\end{array}\right] = \left[\begin{array}{cc}-2 & 2\\0 & 0\end{array}\right]$$$
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.
$$$\lambda = 1$$$
$$$\left[\begin{array}{cc}1 - \lambda & 2\\0 & 3 - \lambda\end{array}\right] = \left[\begin{array}{cc}0 & 2\\0 & 2\end{array}\right]$$$
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.