How Can I Derive a Set of 3x3 Matrices Analogous to the Pauli Matrices?

[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.

 

[Fredrik]

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.

 

[George Jones]

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.

 

[Dahaka14]

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

 

[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!

 

[HallsofIvy]

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

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

 

[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?