rotation matrix

Definition 1.

A rotation matrixMathworldPlanetmath is a (real) orthogonal matrixMathworldPlanetmath whose determinantMathworldPlanetmath is +1. All n×n rotation matrices form a group called the special orthogonal groupMathworldPlanetmath and it is denoted by SO(n).


  1. 1.

    The identity matrixMathworldPlanetmath in n is a rotation matrix.

  2. 2.

    The most general rotation matrix in 2 can be written as


    where θ. Multiplication (from the left) with this matrix rotates a vector (in 2) θ radians in the anti-clockwise direction.


  1. 1.

    Suppose vn is a unit vectorMathworldPlanetmath. Then there exists a rotation matrix R such that Rv=(1,0,,0).

  2. 2.

    In fact, for vn, n3, there are many rotation matrices 𝐑SO(n) such that Rv=(1,0,,0)T. To see this, let f be the mapping f:SO(n-1)SO(n), defined as


    Then for each QSO(n-1), f(Q) maps (1,0,,0)T onto itself. Thus, if R0SO(n) satisfies Rv=(1,0,,0)T, then f(Q)R satisfies the same property for all QSO(n-1).

Title rotation matrix
Canonical name RotationMatrix
Date of creation 2013-03-22 15:03:57
Last modified on 2013-03-22 15:03:57
Owner matte (1858)
Last modified by matte (1858)
Numerical id 17
Author matte (1858)
Entry type Definition
Classification msc 15-00
Synonym rotational matrix
Related topic OrthogonalMatrices
Related topic ExampleOfRotationMatrix
Related topic DecompositionOfOrthogonalOperatorsAsRotationsAndReflections
Related topic DerivationOfRotationMatrixUsingPolarCoordinates
Related topic DerivationOf2DReflectionMatrix
Related topic TransitionToSkewAngledCoordinates