Orthogonal Matrix

45. Orthogonal Matrix

A square matrix $𝑄$ whose transpose is its inverse:

𝑄^{𝑇} 𝑄 = 𝑄 𝑄^{𝑇} = 𝐼

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

𝑞_{𝑖} \cdot 𝑞_{𝑗} = {\begin{cases} 1 if 𝑖 = 𝑗 \\ 0 if 𝑖 \neq 𝑗 \end{cases}

An orthogonal matrix represents a rigid linear transformation of $ℝ^{𝑛}$ : it preserves lengths and angles.

For any vectors $\vec{𝑢}, \vec{𝑣} \in ℝ^{𝑛}$ :

Lengths: $‖ 𝑄 \vec{𝑣} ‖ = ‖ \vec{𝑣} ‖$ (norm preserved)
Angles: $(𝑄 \vec{𝑢}) \cdot (𝑄 \vec{𝑣}) = \vec{𝑢} \cdot \vec{𝑣}$ (dot product preserved)

Determinant: $det (𝑄) = \pm 1$
- $+ 1$ → orientation preserving (a rotation)
- $- 1$ → orientation reversing (includes reflections)
Eigenvalues: all have absolute value $1$ (they sit on the unit circle in $ℂ$ )
Inverse: $𝑄^{- 1} = 𝑄^{𝑇}$ — extremely cheap to invert
Closed under product: $𝑄_{1} 𝑄_{2}$ orthogonal ( $𝑄_{1}, 𝑄_{2}$ orthogonal)
Orthogonal group $𝑂 (𝑛) = {𝑄 \in ℝ^{𝑛 \times 𝑛} : 𝑄^{𝑇} 𝑄 = 𝐼}$
Special orthogonal group $𝑆 𝑂 (𝑛) = {𝑄 \in 𝑂 (𝑛) : det (𝑄) = 1}$ — pure rotations

Example

$2 \times 2$ rotation matrix by angle $𝜃$ — see Rotation Matrix:

𝑅 (𝜃) = [\begin{matrix} cos 𝜃 & - sin 𝜃 \\ sin 𝜃 & cos 𝜃 \end{matrix}]

${𝑅 (𝜃)}^{𝑇} 𝑅 (𝜃) = 𝐼$ , $det = {cos}^{2} 𝜃 + {sin}^{2} 𝜃 = 1$ .