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Consequently, this two-dimensional rigid-body transformation can be viewed geometrically as involving a rotation around the fixed point (53.4352, — 18.3601).

A useful method for describing a rigid-body movement within the framework of a given coordinate system is to use a decomposition based on the matrix logarithm of the transformation matrix. By definition, the matrix logarithm of a matrix has eigenvectors that are identical to those of the original matrix, but all of the eigenvalues are the natural logarithms of the eigenvalues of the original matrix. The inverse matrix function, the matrix exponential, is defined similarly. Although defined here with reference to the eigenvalues and eigenvectors of a matrix, both of these matrix functions can be computed without explicit computation of the corresponding eigenvectors.

For the rigid-body example transformation, the matrix logarithm is of the transformation is

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