Matrix Transformation Visualiser
Enter a 2x2 matrix and see how it transforms the coordinate plane.
Linear Manifold Distortions 🖖
Matrix transformations represent the ultimate linear operators, forcefully warping the Cartesian plane through shear, scaling, and rotation matrices. The determinant explicitly calculates the exact global area scaling factor, while the elusive eigenvectors remain completely immune to rotational deflection under the applied transform.
The columns tell you everything 🖖
A 2×2 matrix is really just instructions for two arrows. Its first column shows where the arrow (1, 0) lands, and its second column shows where (0, 1) lands. Because straight lines stay straight and the grid stays evenly spaced, those two destinations fix where every other point goes. To predict any transformation, just read its columns.
Rotations have no favourite direction 🖖
Stretches and shears keep at least one line pointing its original way — an eigenvector. A pure rotation, though, tilts every direction, so it has no real eigenvectors at all; its eigenvalues are the complex pair cos θ ± i·sin θ. That is exactly the eiθ that multiplies complex numbers — rotating the plane and multiplying by a unit complex number are the same operation.
Example problems
- Rotate 45° - Rotate 45°
- Shear - Shear
- Scale - Scale
- Reflect - Reflect
- Singular - Singular