# Generalized Spin Matrices

1. Dec 16, 2008

### Dahaka14

If I am to try and derive a set of 3x3 matrices analogous to the Pauli matrices, how would I go about doing this? I want to find the basis for all complex 3x3 matrices (analagous to the 3 Pauli matrices and the identity matrix for all complex 2x2 matrices) to expand a complicated matrix into so that I can derive a simpler expression. I've only just taken a basic linear algebra class, so I am not an EXPERT in vector spaces. Can anyone help me with this?

This is NOT homework for me, just pure theory, so I didn't post it in the homework forum.

2. Dec 16, 2008

### Fredrik

Staff Emeritus
I don't know, but the Pauli matrices are easy to find by explicitly calculating the most general form of a traceless self-adjoint ($A=A^\dagger$) 2×2 matrix, so maybe you can start by doing something like that.

3. Dec 16, 2008

### George Jones

Staff Emeritus
I don't know how useful this is for your purposes, but 3x3 generalizations of the Pauli matrices used in physics are the Gell-Mann matrices,

http://en.wikipedia.org/wiki/Gell-Mann_matrices.

4. Dec 16, 2008

### Dahaka14

thanks a lot! that should help me a lot, i will reincarnate this forum if i need further help

5. Jan 2, 2009

### Dahaka14

I am trying right now to decompose a 3x3 matrix into a basis that can be visualized as a 3-D vector. I did this for a 2x2 matrix analogous to this problem, but the 3x3 matrix I have is quite a bit more complicated. I decomposed the 2x2 with the Pauli matrices, and then made it visualizable in 2-D space. Is this process possible with a 3x3 matrix? I have been trying to use the Gell-Mann matrices, but I am not having luck putting it into a 3-D basis. Let me know if you need more information, I just want to be able to do it myself with only a little help. Could you guys please help? Thanks!

6. Jan 2, 2009

### HallsofIvy

Staff Emeritus
You would have to kill it first! Are you sure "reincarnate" is the word you meant?

7. Jan 2, 2009

### Dahaka14

well, the this thread could be considered dead since it is inactive, plus wouldn't restarting under the same name be considered spamming? but this does not answer my question

do u want me to start a new thread?