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Matrix Transformation 2D Tool

Enter a 2x2 matrix and see how it transforms the unit square and basis vectors. The tool computes the determinant, eigenvalues, and shows the transformed coordinate grid.

Transformation Diagram

Te1Te2OriginalTransformed

Matrix Entries

2 x 2 Transformation Matrix

[
]

Results

Determinant (area scale factor)
1
Trace
2
Eigenvalue 1
1
Eigenvalue 2
1
e1 = [1, 0] maps to
[1, 0]
e2 = [0, 1] maps to
[0, 1]

Step-by-Step Calculation

1. Transformation of Basis Vectors

T[xy]=[1001][xy]T\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}
e^1=[10]→[10],e^2=[01]→[01]\hat{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \to \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad \hat{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \to \begin{bmatrix} 0 \\ 1 \end{bmatrix}

2. Determinant (Area Scale Factor)

det⁡(T)=ad−bc\det(T) = ad - bc
det⁡(T)=(1)(1)−(0)(0)\det(T) = (1)(1) - (0)(0)
det⁡(T)=1\det(T) = 1

3. Trace

tr(T)=a+d\text{tr}(T) = a + d
tr(T)=1+1\text{tr}(T) = 1 + 1
tr(T)=2\text{tr}(T) = 2

4. Eigenvalues

λ2−tr(T)λ+det⁡(T)=0\lambda^2 - \text{tr}(T)\lambda + \det(T) = 0
λ2−2λ+1=0\lambda^2 - 2\lambda + 1 = 0
λ1=1,λ2=1\lambda_1 = 1, \quad \lambda_2 = 1

Reference Guide

Linear Transformations

A 2x2 matrix defines a linear transformation of 2D space. Every point (x, y) maps to a new point (x', y') by matrix multiplication.

[x′y′]=[abcd][xy]\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

The Determinant

The determinant tells you how the transformation scales area. A determinant of 2 means areas double. A negative determinant means orientation is reversed (mirror).

det⁡(T)=ad−bc\det(T) = ad - bc

If the determinant is zero, the transformation collapses 2D space down to a line or a single point.

Eigenvalues

An eigenvalue λ\lambda is a scalar such that the transformation only stretches (or flips) a vector without changing its direction.

Tv⃗=λv⃗T\vec{v} = \lambda\vec{v}

Rotations produce complex eigenvalues. Reflections and scaling produce real eigenvalues.

Common Transformations

Rotation by angle θ\theta uses cos⁡θ\cos\theta and sin⁡θ\sin\theta. Scaling uses diagonal entries. Shear uses off-diagonal entries. Reflection flips one axis.

Try the presets to see each transformation type in action.