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

Matrix Entries

2 x 2 Transformation Matrix

[

a

b

c

d

]

Presets

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\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

\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) = (1)(1) - (0)(0)

\det(T) = 1

3. Trace

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

\text{tr}(T) = 1 + 1

\text{tr}(T) = 2

4. Eigenvalues

\lambda^2 - \text{tr}(T)\lambda + \det(T) = 0

\lambda^2 - 2\lambda + 1 = 0

\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.

\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

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

Eigenvalues

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

T\vec{v} = \lambda\vec{v}

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

Common Transformations

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

Try the presets to see each transformation type in action.

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