Can Any Traceless Self-Adjoint 2x2 Matrix Be Expressed Using Pauli Matrices?

[Frank Einstein]

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

 

[blue_leaf77]

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.

 

[The_Duck]

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

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

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.