Consequently, in the two-dimensional case, a real eigenvector with any value other than zero as its final element can be viewed as a point (a/c, b/c) that is mapped by the transformation to the point (k*a/c, k*b/c). It can be shown mathematically that any rigid-body transformation that involves a rotation must have one real eigenvalue with a nonzero final element and that the eigenvalue of this eigenvector must be 1. Consequently, for such rigid-body transformations, the point (a/c, b/c) is mapped back to itself by the transformation.

Using a standard eigenvector routine to compute the eigenvectors of the example transformation shown earlier, gives a = 0.9456, b =-0.3249, and c = 0.0177, so the point that is unchanged by the transformation is (53.4352, — 18.3601), which can be confirmed as follows:

' 53.4352 "

0 0

Post a comment