Re: solving for multiple matrices
"Staffan Langin" <[email protected]>
| Newsgroups | gmane.games.devel.algorithms |
|---|---|
| Message-ID | <[email protected]> |
Hi there Gino, As you most probably already know, Cholesky decomposition can only be applied to a symmetric, PSD matrix. Even if the matrices only contain rotational transformations, I don't believe Y^T * D * Y generally would result in a symmetric, PSD matrix. In particular I don't believe the assertion C = U^T * U and D = V^T * V is generally true for matrices containing rotational transformations, since that implies that C and D are symmetric and PSD. Best regards, Staffan Langin -----Original Message----- From: Gino van den Bergen [mailto:[email protected]] Sent: den 18 september 2009 09:59 To: Game Development Algorithms Subject: Re: [Algorithms] solving for multiple matrices 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 <[email protected]> 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 > [email protected] > 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 [email protected] 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 [email protected] https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list Archives: http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list