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

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