Matrices that represent the projectors

[jalalmalo]

Here is a home assignment, I don't want it to be solved, just help understand what the teacher is after in one specific part. I looked in the course litterature and searched the web and couldn't find anything relevant.

calculate the eigenvalues and eigenvectors of a certain matrix which I don't know how to write down here, but it doesn't matter.

3. Calculate the matrices which represent the projectors onto the eigen-
vectors that you have specifed.

Here is my confusion, what does he mean by projectors onto the eigenvalue. I know that there is a projection matrix that takes one vector and projects it onto a line och a plane, but I guess this is not what the teacher is after.

by the way the eigenvectors happen to be the spin operators in the +y and -y direction after normalisation.

thanx

 

[tiny-tim]

3. Calculate the matrices which represent the projectors onto the eigen-
vectors that you have specifed.

Here is my confusion, what does he mean by projectors onto the eigenvalue

Hi jalalmalo!

The projector is usually the operator that moves everything perpendicularly onto the line in question …

eg the projector onto the (1,0,0,0) vector sends (x,y,z,t) onto (x,0,0,0) …

I don't think it has anything to with the original matrix.

 

[jalalmalo]

Hi jalalmalo!

The projector is usually the operator that moves everything perpendicularly onto the line in question …

eg the projector onto the (1,0,0,0) vector sends (x,y,z,t) onto (x,0,0,0) …

I don't think it has anything to with the original matrix.[/QUOTE

So propably he is after the operator that projects basis vectors (1,0) and (0,1) onto each of my eigenvectors.


thnx once again.

 

[Hurkyl]

So propably he is after the operator that projects basis vectors (1,0) and (0,1) onto each of my eigenvectors.

But that's only true if you're writing things relative to a basis where your eigenvectors are the standard basis vectors.

 

[tiny-tim]

Hi jalalmalo!

So propably he is after the operator that projects basis vectors (1,0) and (0,1) onto each of my eigenvectors.

I'm not sure what you mean by that …

if you mean an operator that projects both of them onto the direction of one eigenvector, and then another operator that projects both of them onto the direction of the other eigenvector, then that's right

(I assume it's two-dimensional)

 

[jalalmalo]

I will attach the assignement.

thnx all for your help

 

[tiny-tim]

Hi jalalmalo!

The projector is usually the operator that moves everything perpendicularly onto the line in question …

so for example the projector onto the (1 0) direction is

1 0
0 0

since it sends any vector (x y) onto (x 0)