# 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 $\left(1 - \lambda\right) \left(3 - \lambda\right)$ (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.

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.