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})

Read more about this topic:  Orthogonal Matrix

Famous quotes containing the word examples:

    “No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
    —André Breton (1896–1966)

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)