# 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 $$$\left(1 - \lambda\right) \left(3 - \lambda\right)$$$ (for steps, see determinant calculator).

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

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

These are the eigenvalues.

Next, find the eigenvectors.

$$$\lambda = 1$$$

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

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.

$$$\lambda = 3$$$

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

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.

## Answer

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

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