Editing Rotation matrix (section)

== Examples ==
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*The {{nowrap|2 × 2}} rotation matrix
::<math> Q = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} </math>
:corresponds to a 90° planar rotation clockwise about the origin.

*The [[transpose matrix|transpose]] of the {{nowrap|2 × 2}} matrix
::<math> M = \begin{bmatrix} 0.936 & 0.352 \\ 0.352 & -0.936 \end{bmatrix} </math>
:is its inverse, but since its determinant is −1, this is not a proper rotation matrix; it is a reflection across the line {{math|11''y'' {{=}} 2''x''}}.

*The {{nowrap|3 × 3}} rotation matrix
::<math> Q = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \frac{\sqrt{3}}{2} & \frac12 \\ 0 & -\frac12 & \frac{\sqrt{3}}{2} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos 30^\circ &  \sin 30^\circ \\ 0 & -\sin 30^\circ  &  \cos 30^\circ \\ \end{bmatrix} </math>
:corresponds to a −30° rotation around the {{mvar|x}}-axis in three-dimensional space.

*The {{nowrap|3 × 3}} rotation matrix
::<math> Q = \begin{bmatrix} 0.36 & 0.48 & -0.80 \\ -0.80 & 0.60 & 0.00 \\ 0.48 & 0.64 & 0.60 \end{bmatrix} </math>
:corresponds to a rotation of approximately −74° around the axis {{nowrap|(−{{sfrac|1|2}},1,1)}} in three-dimensional space.

*The {{nowrap|3 × 3}} [[permutation matrix]]
::<math> P = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} </math>
:is a rotation matrix, as is the matrix of any [[even permutation]], and rotates through 120° about the axis {{math|''x'' {{=}} ''y'' {{=}} ''z''}}.

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*The {{nowrap|3 × 3}} matrix
::<math> M = \begin{bmatrix} 3 & -4 & 1 \\ 5 & 3 & -7 \\ -9 & 2 & 6 \end{bmatrix} </math>
:has determinant +1, but is not orthogonal (its transpose is not its inverse), so it is not a rotation matrix.

*The {{nowrap|4 × 3}} matrix
::<math> M = \begin{bmatrix} 0.5 & -0.1 & 0.7 \\ 0.1 & 0.5 & -0.5 \\ -0.7 & 0.5 & 0.5 \\ -0.5 & -0.7 & -0.1 \end{bmatrix} </math>
:is not square, and so cannot be a rotation matrix; yet {{math|''M''<sup>T</sup>''M''}} yields a {{nowrap|3 × 3}} identity matrix (the columns are orthonormal).

*The {{nowrap|4 × 4}} matrix
::<math> Q = -I = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} </math>
:describes an [[SO(4)#Isoclinic rotations|isoclinic rotation]] in four dimensions, a rotation through equal angles (180°) through two orthogonal planes.

*The {{nowrap|5 × 5}} rotation matrix
::<math> Q = \begin{bmatrix} 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} </math>
:rotates vectors in the plane of the first two coordinate axes 90°, rotates vectors in the plane of the next two axes 180°, and leaves the last coordinate axis unmoved.
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