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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