Similarity Transformation

An Angle-preserving transformation. A similarity transformation has a transformation Matrix ${\hbox{\sf A}}'$ of the form

$${\hbox{\sf A}}' \equiv {\hbox{\sf B}}{\hbox{\sf A}}{\hbox{\sf B}}^{-1}.$$

If ${\hbox{\sf A}}$ is an Antisymmetric Matrix ( $a_{ij} = -a_{ji}$) and ${\hbox{\sf B}}$ is an Orthogonal Matrix, then

$$(bab^{-1})_{ij} = b_{ik}a_{kl}b^{-1}_{lj} = -b_{ik}a_{lk}b^{-1}_{lj} = -{b^\dagger}_{ki} a_{lk} {(b^\dagger)^{-1}}_{jl}$$

$$= -{b^{-1}}_{ki}a_{ki}b_{jl}= b_{jl}a_{lk}b^{-1}_{ki} = -(bab^{-1})_{ji}.$$

Similarity transformations and the concept of Self-Similarity are important foundations of Fractals and Iterated Function Systems.

See also Conformal Transformation

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 83-103, 1991.