I want to know the geometrical significance of eigenvectors and the work done so far on this subject.?

Answer 1

For dynamical systems, eigenvalues and eigenvectors tell you the long term, equilibrium behavior. For a spacecraft, they tell you the modes of vibration of the spacecraft structure so that you can make sure that the accoustic environment of your launch vehicle won't shake it apart.

Many, many examples.

Geometrically, they show you special directions (invariant subspaces) of your vectors space that are simply stretched or shrunk, the direction remains the same. Used for the projections that you see in advertising where it looks like a 3 dim figure is rotating.

Answer 2

The geometrical meaning of eigenvectors depends on what the matrix or linear operator describes. For example, when a symmetric matrix is used to define a quadratic curve or surface, eigenvectors are the principal axes. When the matrix defines a linear operator in a space, eigenvectors are the invariant directions (those which the operator merely stretches); this helps with solving systems of linear differential equations.
There are many other interpretations in physics, including classical and quantum mechanics.

Answer 3

In a homohorphic vector function, the eigenvectors 'point' in the same direction before and after being acted on by the function and differ only in their magnitude (the constant of proportionality being teh 'eigenvalue' associated with the eigen vector)

Typing 'eigenvector' into a search engine netted me 332,000 hits. You go wade through them ☺


Doug

Answer 4

the geometrical significance of an eigenvector is that it is invariant under the function for wich it is an eigenvector.

Answer 5

in case your asking no be counted if and Irish guy would be indignant it somewhat relies upon on the guy. working example my dads Irish and he tells us(me and siblings) way worse Irish jokes i think of its a fantastically reliable comic tale anyhow