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# FP4 geometrical interpretation of eigenvectors Watch

1. 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
2. (Original post by Vernish)
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
3. 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.
4. (Original post by SherlockHolmes)
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.
5. (Original post by Vernish)
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
If M is the identity matrix then every point of the plane is invariant. If M is a multiple of the identity matrix then every point of the plane will be mapped onto a point of the plane
6. (Original post by SherlockHolmes)
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?
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:

where v is a vector and a is a multiple of v.
8. (Original post by SherlockHolmes)
An eigenvector A, must obey the following equation:

where v is a vector and a is a multiple of v.
Thank you Mr Holmes, you help has been invaluable.

Repped.
9. (Original post by Vernish)
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

If you have any 2 eigenvectors (say with eigenvalues ), these span a plane in . Any point in this plane can be written in the form for some real numbers . So if is a point in the plane,

which is a point in the plane (since it is a linear combination of ).
10. (Original post by james22)
I'm assuming that this is for a 3x3 matrix (although it works for any higher dimension as well), call is

If you have any 2 eigenvectors (say with eigenvalues ), these span a plane in . Any point in this plane can be written in the form for some real numbers . So if is a point in the plane,

which is a point in the plane (since it is a linear combination of ).
Thank you, I think that's made sense.

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