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Matrix : linear :: (?) : nonlinear

Started by kSquared 1 reply 850+ views
Original Post
kSquared
kSquared
Consider the following statement: "For every linear transformation, there's exactly one corresponding matrix, and for every matrix there is a unique linear transformation." Is there an equivalent for nonlinear transformations? That is, can this blank be filled in? "For every nonlinear transformation, there's exactly one corresponding [ ? ], and for every [ ? ] there is a unique nonlinear transformation."
- k2 "Choose a job you love, and you'll never have to work a day in your life." — Confucius"Logic will get you from A to B. Imagination will get you everywhere." — Albert Einstein"Money is the most egalitarian force in society. It confers power on whoever holds it." — Roger Starr{General Programming Forum FAQ<
alvaro
alvaro
Linear transformations are completely determined by the image of the vectors of a base (basis?). That's why you can describe them with as little information as a single matrix.

If by non-linear you mean an arbitrary mapping from R^n to R^m, there are too many such functions to characterize them with any object that is any simpler.

If you are interested in algebraic or analytic transformations, then there is some hope (a few polynomials or a few power series will do the job).
Dmytry
Dmytry
Exactly as alvaro said.

For example, image defines usually nonlinear(piecewise linear, for example) 2D->3D mapping, where resulting coordinate is color of pixel...

Some specific nonlinear transformations, for example perspective transformation can be somehow storen in matrix. Perspective transform is given by ratio of 2 linear transforms, so matrix is used to linearly transform some vector that is then divided by w component, so we get ratio of 2 linear transforms...

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