# 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

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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 $$\lambda^{2} - 4 \lambda + 3$$$ (for steps, see determinant calculator).

Solve the equation $$\lambda^{2} - 4 \lambda + 3 = 0$$$. The roots are $$\lambda_{1} = 3$$$, $$\lambda_{2} = 1$$$(for steps, see equation solver). These are the eigenvalues. Next, find the eigenvectors. 1. $$\lambda = 3$$$

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

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

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. ## Answer Eigenvalue: $$3$$$A, multiplicity: $$1$$$A, eigenvector: $$\left[\begin{array}{c}1\\1\end{array}\right]$$$A

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