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# Eigenvectors/Eigenvalues

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Hi all, Just a quickie: what can eigenvectors and/or eigenvalues be used for in the context of game physics/graphics/whatever? Is there any real purpose to their existence? Tom L

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Finding the axis of rotation of a rotation matrix is one. I don't know if it is of particular use in games, but you can use them to find powers of matrices. Considering some of the things people do with graphics I would imagine it has some uses well over my head.

[edited by - LilBudyWizer on October 19, 2002 1:57:59 PM]

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Eigenvalues/vectors are quite often seen in analysis of the convergence/error properties of numerical methods. So, if you wish to know your numerical methods more deeply, study eigenvectors.

To freely quote Shewchuck''s great name monster paper "Introduction to the conjugate gradient method without the agonizing pain": one of the most important skills in linear algebra is to be able (at least in your head) to compose a vector whose properties are unknown as a linear combination of vectors whose properties are known. For example, any symmetric NxN matrix has N linearly independent eigenvectors, so any vector in N space can be intuitively expressed as a linear combination of these.

I''ve hard they have some applications with signal processing too, e.g. computer vision.

- Mikko Kauppila

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Finding Eigenvalues/eigenvectors is important in an inumerable number of fields. Some examples that relate to game development:

Pattern recognition, signal processing, differential equations (classical and stochastic), information theory, kinematics, linear algebra...

Certainly eigenvalues and eigenvectors seem rather esoteric when you''re learning them as part of an introductory course on linear algebra, but they are certainly useful (as a tool) later in your mathematical life.

Cheers,

Timkin

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Eigenvalues/vectors can be used to find the principle axes of inertia and the principle inertias of a body, which can sometimes simplify physics calculations, such as rigid-body dynamics. If you distribute mass equally to all nodes in a mesh, the principle inertia axes also might represent a good guess at the minimal oriented bounding box (OBB) for the object. I think those are two occasionally useful uses for game development, .

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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