2D Matrix Transformation Visualizer
Explore how linear and affine transformations act on objects in the plane. Animate rotations, reflections, scaling, shear, and projections. Decompose any matrix via SVD and compose sequences of transformations.
Linear Transformations
A 2×2 matrix A defines a linear transformation that maps every point x in the plane to Ax.
- Rotation – rotates points by a given angle about the origin
- Reflection – reflects points across a line through the origin
- Scaling – stretches or compresses along the coordinate axes
- Shear – slides points parallel to an axis proportionally to their distance from it
- Projection – collapses points onto a line through the origin
SVD Decomposition
Every matrix can be decomposed as A = U Σ VT, revealing the geometry: a rotation (VT), a scaling (Σ), and another rotation (U). Use the Decompose button to animate each step.
Composition
Add multiple transformations to the queue and play them in sequence. The resulting matrix is the product of all queued matrices, applied right to left.
Transformation
°
Translation (affine shift)
Object & Display
Animation
Transform
Animates the current matrix transformation on the grid and shape.
Resets the animation and restores the identity matrix view.
Animates the SVD decomposition step-by-step: rotate, scale, rotate.
1.0×
Composition Queue
Build a sequence of transformations. The product matrix is applied right-to-left.
Adds the current transformation matrix to the composition queue.
Removes all transformations from the queue.
Animates the composed transformation sequence step by step.
e₁ → Ae₁
e₂ → Ae₂
Unit circle
Ghost (original)