Re: [Algorithms] solving for multiple matrices

[Bill B. <wb...@gm...>]

> -----Original Message-----
> From: Andras Balogh [mailto:and...@gm...]
> Sent: Thursday, September 17, 2009 12:38 PM
> To: gda...@li...
> Subject: [Algorithms] solving for multiple matrices
>
> Hi, I have a chain of transformations with multiple unknown (but fixed!) transforms. What I do know is the end result transformation and some of the transformations in between, and I know these for multiple frames. So
>  from here, I'd like to compute the unknowns. Here it is in more formal
> version:
>
> I'd like to find 2 unknown matrices X and Y. I have 4 known matrices A1, A2, B1 and B2, and also know this:
> A1 = X * B1 * Y
> A2 = X * B2 * Y
>
> I can compute X from the first equation:
> X = A1 * Y^-1 * B1^-1
>
> And substitute it into the second:
> A2 = A1 * Y^-1 * B1^-1 * B2 * y
>
> Assigning:
> C = A1^-1 * A2
> D = B1^-1 * B2
>
> Then it becomes:
> C = Y^-1 * D * Y
>
> Now, how do I solve this for Y? This form lookes strangely familiar, but I cannot figure out what to do from here (wish I knew how to Google this ;).
> Hopefully there's an analytic solution to this. Any ideas?
>
> Thanks,
>
>
>
> Andras

It has the form of a similarity transformation, if that helps.
http://mathworld.wolfram.com/SimilarityTransformation.html

Similarity transforms preserve eigenvalues and consequently the trace
and determinant of the matrix applied to, so if C and D don't have the
same trace and determinant, then your Y doesn't exist.

--bb