Rotation Matrices

\begin{bmatrix} cos(\theta && - sin(\theta) \\ sin(\theta) && cos(\theta) \end{bmatrix}

\begin{bmatrix} cos(\theta && - sin(\theta) && 0 \\ sin(\theta) && cos(\theta) \\ 0 && 0 && 1 \end{bmatrix}

A rotation matrix is not a tensor because it is specific to the old coordinate system and the new coordinate system

Appendix A

Work below is done in two dimensions. It might make a good homework problem for you to show the same in three dimensions.

Work below makes use of the idea that cosine is an Even Function:

cos(\theta) = cos(- \theta)

…and also that sine is Odd Function:

sin(\theta)= - sin(- \theta)

We can show that if we take the transpose of this rotation Matrix and then take the inverse of that, we get the original. We can also show that if we take the inverse and then take the transpose of that, we get the original.

  • (A^{-1})^T = A
  • (A^T)^{-1} = A

A = \begin{bmatrix} cos(\theta && - sin(\theta) \\ sin(\theta) && cos(\theta) \end{bmatrix}

A^T = \begin{bmatrix} cos(\theta && sin(\theta) \\ - sin(\theta) && cos(\theta) \end{bmatrix}

Appendix B

In four dimensions (w,x,y,z) a rotation in the xy plane is rotating about two axes, w and z.