Math 2.0

Linear and multilinear algebra; matrix theory Example

Rotation matrix

Builds on Orthogonal matrixSpecial orthogonal group

Example

Let θ∈ℝ\theta \in \R. The rotation matrix of angle θ\theta is R(θ)=(cos⁡θ−sin⁡θsin⁡θcos⁡θ).R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}. The map x↦R(θ)xx \mapsto R(\theta) x rotates the plane ℝ2\R^2 about the origin by the angle θ\theta, counterclockwise. It is an orthogonal matrix with determinant cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1, so it is an element of SO(2)SO(2).

Remarks

Rotations add their angles: R(α)R(β)=R(α+β)R(\alpha) R(\beta) = R(\alpha + \beta). So R(θ)−1=R(−θ)=R(θ)TR(\theta)^{-1} = R(-\theta) = R(\theta)^T. For θ\theta not a multiple of π\pi, R(θ)R(\theta) has no real eigenvalue; its complex eigenvalues are e±iθe^{\pm i\theta}. Every element of SO(2)SO(2) is R(θ)R(\theta) for some θ\theta.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectRotation matrix, example, no evidence levelRotation matrixexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Transpose definition
  6. Linear map definition
  7. Orthogonal matrix definition
  8. Multilinear map definition
  9. Orthogonal group object
  10. Alternating multilinear map definition
  11. Determinant definition
  12. Special orthogonal group object
  13. Rotation matrix example

Further reading

  1. Web https://en.wikipedia.org/wiki/Rotation_matrix, In two dimensions
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WritingW2 usable
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Instance ofSpecial orthogonal group
UsesOrthogonal matrix, Special orthogonal group
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