Orthogonal Matrix - Examples

Examples

Below are a few examples of small orthogonal matrices and possible interpretations.

  • \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad (\text{identity transformation})

An instance of a 2×2 rotation matrix:

  • R(16.26^\circ) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix} = \begin{bmatrix} 0.96 & -0.28 \\ 0.28 & \;\;\,0.96 \\ \end{bmatrix} \qquad (\text{rotation by }16.26^\circ )
  • \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} \qquad (\text{reflection across }x\text{-axis})
  • \begin{bmatrix} 0 & -0.80 & -0.60 \\ 0.80 & -0.36 & \;\;\,0.48 \\ 0.60 & \;\;\,0.48 & -0.64 \end{bmatrix} \qquad \left( \begin{align}&\text{rotoinversion:} \\&\text{axis }(0,-3/5,4/5),\text{ angle }90^{\circ}\end{align}\right)
  • \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \qquad (\text{permutation of coordinate axes})

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