Transformation Matrices Calculator

Transformation Matrix:

Understanding Matrix Transformations

Basic Principles

2D Transformation Matrix:

[a b]

[c d]

Rotation Matrix

  • R(θ) = [cos θ -sin θ]
  • [sin θ cos θ]
  • Preserves shape and size
  • Determinant = 1
  • Orthogonal matrix

Scaling Matrix

  • S(sx,sy) = [sx 0]
  • [0 sy]
  • Uniform vs non-uniform
  • Area factor = sx×sy
  • Preserves angles if sx=sy

Reflection Matrix

  • X-axis: [1 0]
  • [0 -1]
  • Y-axis: [-1 0]
  • [0 1]
  • Determinant = -1

Advanced Concepts

Matrix Properties

  • Eigenvalues and eigenvectors
  • Singular value decomposition
  • Jordan canonical form
  • Matrix exponential
  • Characteristic polynomial

Composition Rules

  • Matrix multiplication
  • Order dependence
  • Inverse transformations
  • Group properties
  • Decomposition theorems

Special Cases

  • Projective transformations
  • Affine transformations
  • Similarity transformations
  • Conformal mappings
  • Isometries

Applications

Computer Graphics

  • 2D/3D rendering
  • Animation systems
  • Game development
  • Image processing
  • Virtual reality

Engineering

  • Robotics kinematics
  • Computer vision
  • Signal processing
  • Control systems
  • Structural analysis

Scientific Computing

  • Quantum mechanics
  • Data visualization
  • Statistical analysis
  • Machine learning
  • Numerical methods