# Matrices that represent the projectors

1. Aug 19, 2009

### 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

2. Aug 19, 2009

### tiny-tim

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.

3. Aug 20, 2009

### jalalmalo

4. Aug 20, 2009

### Hurkyl

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

5. Aug 20, 2009

### tiny-tim

Hi jalalmalo!
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)

6. Aug 20, 2009

### jalalmalo

I will attach the assignement.

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7. Aug 21, 2009

### 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

Code (Text):
1 0
0 0
since it sends any vector (x y) onto (x 0)