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

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

See also Conformal Transformation


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

© 1996-9 Eric W. Weisstein