Let us assume A ,B and C are known affine transformation matrices in homogeneous 2D space.If it should happen that C = A m B n or some unknown m,n, is there a way to detect this short of trying all the combinations? If one of A,B is an identity matrix this can be determined by examining the eigenvectors, but is there an equivalent way of doing this for a product such as this?

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Let us assume A ,B and C are known affine transformation matrices in homogeneous 2D space.If it should happen that   C  =      A    m        B    n   or some unknown m,n, is there a way to detect this short of trying all the combinations? If one of A,B is an identity matrix this can be determined by examining the eigenvectors, but is there an equivalent way of doing this for a product such as this?

soymmernenx · Accepted Answer

If   A and   B commute, you can use the matrix logarithm! Sure, it&#039;s not unique but solutions only vary by integer multiples of   2  π  i you can check by hand whether   log  ⁡  C looks like  log  ⁡  (  C  )  =  q  2  π  i  I  +  n  log  ⁡  (  A  )  +  m  log  ⁡  (  A  )for integers   q  ,  n  ,  m. Each entry in   log  ⁡  (  C  ) corresponds to a linear equation q,n,m must solve. It&#039;s complex, but that doesn&#039;t matter. It&#039;s now a linear system.Here&#039;s why commuting matters. In the real setting,   log  ⁡  (  a  b  )  =  log  ⁡  (  a  )  +  log  ⁡  (  b  ) is always true. However, in the matrix setting addition does commute but multiplication may not. It turns out that  log  ⁡  (  A  B  )  =  log  ⁡  (  A  )  +  log  ⁡  (  B  )    mod    2  π  i  Iis only guaranteed for   A and   B which commute. So while you can do the computation described above for   A and   B which don&#039;t commute, it&#039;ll just give you garbage. The equation   C  =      A    n        B    m   need not have any relation with   log  ⁡  (  C  )  =  q  2  π  i  I  +  n  log  ⁡  (  A  )  +  m  log  ⁡  (  A  ).Computing logarithms is not too hard numerically. Just use the Taylor series. Computing them by hand is not too bad if   A or   B is diagonalizable.If you go the numeric route, just be sure to check that the candidate   n and   m actually work. The matrix log can be a little sensitive near the edges of it&#039;s interval of convergence.I know that this   A and   B commuting is a fairly special case, but you gotta start somewhere. I&#039;ll keep thinking of what to do in the more general case.

Let us assume A ,B and C are known affine transformation matrices in homogeneous 2D space. If it sh

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