2×2 Matrix Visualiser
Change the entries of the matrix and hit enter to update the transformed image of Lena.
\[ \mathbf M=\begin{pmatrix} \FormInput[2][matrix-entry][1]{a} & \FormInput[2][matrix-entry][0]{b}\\ \FormInput[2][matrix-entry][0]{c} & \FormInput[2][matrix-entry][1]{d} \end{pmatrix} \]Determinant: \(\det(\mathbf M)={}\)\(\), eigenvalues: \(\lambda={}\) \(\), \(\). The corresponding eigenvectors (if they exist) are illustrated in the image in red and blue respectively.
Here are some examples of matrix transformations.
| Transformation | Matrix | Try it (enter your parameter and hit enter) |
|---|---|---|
| Rotation by angle \(\theta\) | \(\begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}\) | \(\theta\)= |
| Reflection in line $y=x\tan\theta$ | \(\begin{pmatrix}\cos2\theta & \sin2\theta\\\sin2\theta & -\cos2\theta\end{pmatrix}\) | \(\theta\)= |
| Uniform scaling by factor \(k\) | \(k\mathbf I=\begin{pmatrix}k & 0\\0 & k\end{pmatrix}\) | \(k\)= |
| Shear parallel to \(x\)-axis | \(\begin{pmatrix}1 & k\\0 & 1\end{pmatrix}\) | \(k\)= |
| Shear parallel to \(y\)-axis | \(\begin{pmatrix}1 & 0\\k & 1\end{pmatrix}\) | \(k\)= |
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