The three matrices to the right in this decomposition describe anisotropic rescaling within a frame defined by a set of axes oriented 54.9232° counterclockwise to the coordinate axes. The matrix on the left describes a rigid-body transformation of this reference frame.
The matrix logarithm can be used to decompose an affine transformation, but the geometric insights provided are less usefUl than those provided by singular value decomposition. Instantaneous elementary shears replace instantaneous elementary rotations in a general affine transformation. Figure 8 shows elementary two-dimensional shears along each of the two axes and illustrates that rotations can be viewed as a balanced form of shears in the same way that global rescaling can be viewed as a balanced form of anisotropic rescaling.
For the numerical example, the matrix logarithm is
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