>

**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.