= Rotation Matrix =

A '''rotation matrix''' represents the linear transformation of rotation.

<<TableOfContents>>

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== Description ==

A rotation matrix is defined for a specific number of dimensions. In ''R^2^'', a rotation matrix is:

{{attachment:rot2.svg}}

where ''θ'' is the angle of counter-clockwise rotation. Note that a negative ''θ'' represents clockwise rotation.

As an example, a counter-clockwise rotation of 90 degrees (or ''π/2'' radians) is represented by:

{{attachment:rotex.svg}}

In ''R^3^'', there are separate rotation matrices for each dimension.

{{attachment:rot3.svg}}

where ''α'', ''β'', and ''γ'' represent '''yaw''', '''pitch''', and '''roll''' in the Z, Y, and X dimensions respectively. A complete rotation matrix in three dimensions can then be calculated as '''''R''' = '''R''',,z,,(α)'''R''',,y,,(β)'''R''',,x,,(γ)''.



=== Properties ===

A rotation matrix is always [[LinearAlgebra/Orthogonality#Orthonormality|orthonormal]]. It follows that '''''R'''^T^ = '''R'''^-1^''.

The [[LinearAlgebra/Determinant|determinant]] of a pure rotation matrix, i.e., one that rotates and does not stretch, is either 1 or -1.



=== Decomposition of Rotation and Stretching ===

A 2-dimensional matrix that rotates and stretches can be decomposed between the two components.

Consider the example matrix {{attachment:ab.svg}}:
 * The eigenvalues (''λ'') are ''a + bi'' and ''a - bi''
 * Following from the formula for [[Calculus/ComplexVector#Distance|distance of complex vectors]], the stretching factor is ''||λ|| = √(a^2^ + b^2^)''.

Therefore the decomposition is:

{{attachment:decom.svg}}

where:
 * ''a = ||λ|| cos(θ)'' and ''b = ||λ|| sin(θ)'', or
 * ''θ = tan^-1^(b/a)''

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== Eigenvectors ==

A matrix that rotates will always have [[Calculus/ComplexVector|complex]] [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]]. In fact, a complex eigenpair means that the transformation involves rotation.

Consider the 2-dimensional case again. There clearly cannot be ''any'' vectors which do not change direction through rotation.

Consider now the 3-dimensional rotation matrix for rotation around the Z axis. This is fundamentally the same rotation. But now all vectors along the axis of rotation are unchanged by rotation.



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