Question about Pauli Matrices

  • #1

Main Question or Discussion Point

Hi everybody, a teacher of mine has told me that any complex, self adjoint matrix 2*2 which trace is zero can be written as a linear combination of the pauli matrices.
I want to prove that, but I haven't been able to.
Please, could somebody point me a book where it is proven, or tell me how to do it?

Thanks for reading
 

Answers and Replies

  • #2
blue_leaf77
Science Advisor
Homework Helper
2,629
784
Just try to form arbitrary linear combination of Pauli matrices and see if the resulting matrix complies with the requirement of being called self-adjoint and has zero trace.
$$A = c_1\sigma_1 + c_2\sigma_2 + c_3\sigma_3$$
where the ##c##'s are real constants.
 
  • #3
1,006
104
Hi everybody, a teacher of mine has told me that any complex, self adjoint matrix 2*2 which trace is zero can be written as a linear combination of the pauli matrices.
I want to prove that, but I haven't been able to.
Please, could somebody point me a book where it is proven, or tell me how to do it?

Thanks for reading
Write down a general ##2 \times 2## matrix as

[tex]\left(\begin{array}{cc} a+bi & c+di \\ e + fi & g + hi \end{array} \right)[/tex]

Now require the matrix to be self-adjoint and traceless. What constraints does this put on ##a,b,\ldots,h##? Try to see how the resulting matrix can be written as a linear combination of the three Pauli matrices.
 

Related Threads for: Question about Pauli Matrices

Replies
4
Views
4K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
7
Views
9K
  • Last Post
Replies
1
Views
279
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
7
Views
5K
  • Last Post
Replies
10
Views
3K
Top