Re: [Algorithms] solving for multiple matrices
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Re: [Algorithms] solving for multiple matrices
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From: Andras B. <and...@gm...> - 2009-09-18 14:52:18
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Hmm, I just looked up Cholesky decomposition, and it says you can only do that if the matrix is symmetric and positive definite. Since the original definition of C = A1^-1 * A2, where A1 and A2 both contain arbitrary translation and rotation, I don't think C will be symmetric. Or am I missing something? Andras On Fri, 18 Sep 2009 01:58:41 -0600, Gino van den Bergen <gin...@gm...> wrote: > If Y can be considered a rotation then Y is orthogonal and thus Y^-1 = > Y^T, in which case this equation can be solved through Cholesky > decomposition: > > For C = Y^T * D * Y, let's decompose > > C = U^T * U, and > D = V^T * V > > then > > U^T U = Y^T * V^T * V * Y > = (V * Y) ^T * (V * Y) > > this gives U = V * Y > > and thus Y = V^-1 * U > > Basically you are taking the square root of a matrix. > > Hope this helps, > > Gino > > > > Andras Balogh wrote: >> Both X and Y matrices represent a simple translation and rotation. For >> the >> case of the Y matrix, the translation part will likely to be very small, >> so I could probably pretend it's only rotation. >> >> What I would really like though, is to find a solution, where I could >> use >> more than two equations, eg: >> A1 = X * B1 * Y >> A2 = X * B2 * Y >> A3 = X * B3 * Y >> ... >> An = X * Bn * Y >> >> And then compute a least squares solution from this over-constrained >> system. >> BTW, when I said that I'm looking for an analytical solution, I just >> meant >> something that is not based on an iterative approach. As long as I can >> get >> to a part where I have to solve a large system of over-constrained >> linear >> equations, I'm home. Unfortunately, I don't know how to make this >> linear.. >> >> >> Andras >> >> >> >> >> On Thu, 17 Sep 2009 16:32:25 -0600, Jon Watte <jw...@gm...> wrote: >> >> >>> Andras Balogh wrote: >>> >>>> 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? >>>> >>>> >>>> >>> That's the formula for applying a rotation in the reference frame of >>> another rotation. >>> >>> Do you know anything more about these matrices than that they are >>> matrices? Are they supposed to contain no scale? No translation? If you >>> can formulate them as quaternions, writing out the analytical answer is >>> a lot simpler :-) >>> >>> Sincerely, >>> >>> jw >>> >>> >>> >>> >> >> >> >> ------------------------------------------------------------------------------ >> Come build with us! The BlackBerry® Developer Conference in SF, CA >> is the only developer event you need to attend this year. Jumpstart your >> developing skills, take BlackBerry mobile applications to market and >> stay >> ahead of the curve. Join us from November 9-12, 2009. Register >> now! >> http://p.sf.net/sfu/devconf >> _______________________________________________ >> GDAlgorithms-list mailing list >> GDA...@li... >> https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list >> Archives: >> http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list >> > > > ------------------------------------------------------------------------------ > Come build with us! The BlackBerry® Developer Conference in SF, CA > is the only developer event you need to attend this year. Jumpstart your > developing skills, take BlackBerry mobile applications to market and stay > ahead of the curve. Join us from November 9-12, 2009. Register > now! > http://p.sf.net/sfu/devconf > _______________________________________________ > GDAlgorithms-list mailing list > GDA...@li... > https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list > Archives: > http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list |
