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z9u2K

What the.... (eigenvectors)

6 posts in this topic

Why, oh why, these collision detection algorithms always use math stuff I don''t know? Is there some kind of a guild that invents that stuff? Nevermind, To the point, I''ve recently started to read (I repeate, READ, that doesn''t mean I understand what I''m reading) the SIGGRAPH 96'' document that describes the OBBTree algorithm used in the RAPID library (from NCU). Everything was OK till I got to the 3rd page... WHAT THE HELL IS EIGENVECTORS AND WHAT ARE THEY DOING IN THAT DOCUMENT??? I''v spent 3 days and couldn''t find any normal document that gives a complete tutorial on these eigenvectors. all I came up with was that they are vectors that doesn''t change thire direction whene they are multiplied by a matrix. I don''t know about you guys, but with this explenation I couldn''t understand EXACLLY what are eigenvectors... HELP!!! I''m despret!!! I can''t build that OBBTree without these vectors!!!
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While I have no idea what Eigenvectors are either, I think this might help you: http://www.math.hmc.edu/calculus/tutorials/eigenstuff/
I just found it by typing in "tutorial eigenvectors" (without the quotes) at www.google.com. You should really try to find stuff at google first and ask questions later.

Dirk "Scarab" Gerrits

''If knowledge is power, than to be unknown is to be unconquerable.''
- Unknown Romulan Centurion
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> WHAT THE HELL IS EIGENVECTORS AND WHAT ARE THEY DOING IN THAT DOCUMENT???
> I''v spent 3 days and couldn''t find any normal document that gives a complete
> tutorial on these eigenvectors.

> all I came up with was that they are vectors that doesn''t change thire
> direction whene they are multiplied by a matrix.

That''s it, i.e. that''s all that they are. The other questions are:

1) Why are they interesting/useful ?
They are interesting because they are one way of extracting useful geometric information from a matrix. E.g. in 3D we can prove that every non-identity 3x3 rotation matrix has exactly one eigenvector, from which we know all rotations are given by a rotation about a fixed axis, which is just the eigenvector. And a knowledge of eignenvectors gives a way to determine the axis. Because eigenvectors are defined for all matrices it gives a way to analyse rotations into 4 and higher dimensions.

2) How to work them out ?
The algebraic method is strightforward rather involved, and is difficult to reproduce in HTML. But it''s also possible to deduce them geometrically from the properties of the object described by the matrix.

E.g one of their main applications in dynamics is the moment of inertia tensor. This is symmetric 3x3 matrix describing the moemnt of inertia of a body, and it''s possible to prove that any such matrix has 3 orthogonal eignenvectors. If these are chosen as the axes for calculating the moment of inertia tensor it will be a diagonal matrix, and for this to happen the axes will often lie along axes/planes of symmetry of the object. I.e. the eigenvectors can often just be identified with the axes of symmetry of the object.
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an eigen vector/scalar is this :

suppose you have a transformation T:V->V
then , you will always have a vector - v ( and a matching scalar value - l { called , eigen scalar })so that :
T(v)=l*v

if you represent the transfromation as a matrix A=[T] then :
A*v=l*v

thats all ( not SO complicated right )

some interesting facts about eigen vectors :
1) if you have n eigen vectors :v1,...,vn matching n DIFFERENT eigen scalars , then , v1,...,vn form a base for V ( in most 3d applications V=RxRxR )

2) to find the eigen scalars ( then you can easily find the vectors ) you have to solve the polinom det(A-I*x)=0 ( you do know how to find a determinanta for a matrix , do you ?)

i hope i helped you and didnt just confuse you more .

the rubber-hound
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I think you can use eigenvectors to solve systems of differential equations as well, could be useful for 2D or simple 3D physics simulations.

Magmai Kai Holmlor
- Not For Rent
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quote:
Original post by Magmai Kai Holmlor
I think you can use eigenvectors to solve systems of differential equations as well, could be useful for 2D or simple 3D physics simulations.



Actually, the eigenvectors appear in the solution of such kinematic systems. Eigenvalue methods can be used to find such solutions.

Timkin
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Thanks a lot guys! you''ve helped me alright! Now I can continue(trying) to build my OBBTree!!!
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