Orthonormal Matrix

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An Orthonormal Matrix, [math]\displaystyle{ Q_t }[/math] is an orthogonal matrix (with mutually orthogonal unit vectors) that is equal to its inverse, that is [math]\displaystyle{ Q^\mathrm{T}=Q^{-1} }[/math] .



References

2015

2013


  • http://en.wikipedia.org/wiki/Orthogonal_matrix#Examples
    • Below are a few examples of small orthogonal matrices and possible interpretations.
      • [math]\displaystyle{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad (\text{identity transformation}) }[/math]
      • An instance of a 2×2 rotation matrix: [math]\displaystyle{ 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 ) }[/math]
      • [math]\displaystyle{ \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} \qquad (\text{reflection across }x\text{-axis}) }[/math]
      • [math]\displaystyle{ \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) }[/math]

        ***[math]\displaystyle{ \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}) }[/math]