45. Orthogonal Matrix

A square matrix whose transpose is its inverse:

Equivalently, the columns (and rows) of form an orthonormal set — each is a unit vector and they are mutually orthogonal:

45.1. Geometric meaning

An orthogonal matrix represents a rigid linear transformation of : it preserves lengths and angles.

For any vectors :

• Lengths: (norm preserved)
• Angles: (dot product preserved)

45.2. Properties

• Determinant:

  • → orientation preserving (a rotation)
  • → orientation reversing (includes reflections)
• Eigenvalues: all have absolute value (they sit on the unit circle in )
• Inverse: — extremely cheap to invert
• Closed under product: orthogonal ( orthogonal)
• Orthogonal group
• Special orthogonal group — pure rotations

45.3. Where they show up

• QR decomposition: — orthogonal × upper triangular
• SVD: — both and are orthogonal
• Spectral theorem: real symmetric matrices diagonalize via orthogonal
• Gram–Schmidt: produces orthonormal columns from any basis
• Coordinate frame changes in robotics, computer graphics, physics