FP4 geometrical interpretation of eigenvectors

Hi

I was working with eigenvectors of M representing invariant lines of transformations with matrix M.

Am pastpapering, have come across the interpretation being a plane of invariant points. Please could someone explain to me when, and why, this happens? Is it possibly linked to there being a repeated eigenvalue?

Cheers

Eigenvectors indicate the direction which remain invariant and the eigenvalue determines the length.

If the eigenvalue is 1 it means that the points on the invariant line are not stretched so you will have a line of invariant points.

Hi

I was working with eigenvectors of M representing invariant lines of transformations with matrix M.

Am pastpapering, have come across the interpretation being a plane of invariant points. Please could someone explain to me when, and why, this happens? Is it possibly linked to there being a repeated eigenvalue?

Cheers

Eigenvectors indicate the direction which remain invariant and the eigenvalue determines the length.

If the eigenvalue is 1 it means that the points on the invariant line are not stretched so you will have a line of invariant points.

Hi Mr Holmes, I was looking at a question about the eigenvectors of a matrix and was a bit confused about what this actually meant. Could you clarify this for me please?

An eigenvector A, must obey the following equation:

$Av=\lambda v$

where v is a vector and a $\lambda$ is a multiple of v.

Hi

I was working with eigenvectors of M representing invariant lines of transformations with matrix M.

Am pastpapering, have come across the interpretation being a plane of invariant points. Please could someone explain to me when, and why, this happens? Is it possibly linked to there being a repeated eigenvalue?

Cheers

I'm assuming that this is for a 3x3 matrix (although it works for any higher dimension as well), call is $M$

If you have any 2 eigenvectors (say $u,v$ with eigenvalues $p,q$), these span a plane in $\mathbb{R}^3$. Any point in this plane can be written in the form $au+bv$ for some real numbers $a,b$. So if $w$ is a point in the plane,

$Mw=M(au+bv)=apu+bqv$ which is a point in the plane (since it is a linear combination of $u,v$).