Decompose complex matrix transformation into a series of simple transformations?

math vector matrix transform decomposition
Sergey Ilinsky picture Sergey Ilinsky · Jul 24, 2009 · Viewed 7.6k times · Source

I wonder if it is possible (and if it is then how) to re-present an arbitrary M3 matrix transformation as a sequence of simpler transformations (such as translate, scale, skew, rotate)

In other words: how to calculate MTranslate, MScale, MRotate, MSkew matrices from the MComplex so that the following equation would be true:

MComplex = MTranslate * MScale * MRotate * MSkew (or in an other order)

Answer

Jason S picture Jason S · Jul 24, 2009

Singular Value Decomposition (see also this blog and this PDF). It turns an arbitrary matrix into a composition of 3 matrices: orthogonal + diagonal + orthogonal. The orthogonal matrices are rotation matrices; the diagonal matrix represents skewing along the primary axes = scaling.

The translation throws a monkey wrench into the game, but what you should do is take out the translation part of the matrix so you have a 3x3 matrix, run SVD on that to give you the rotation+skewing, then add the translation part back in. That way you'll have a rotation + scale + rotation + translate composition of 4 matrices. It's probably possible to do this in 3 matrices (rotation + scaling along some set of axes + translation) but I'm not sure exactly how... maybe a QR decomposition (Q = orthogonal = rotation, but I'm not sure if the R is skew-only or has a rotational part.)

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