quantum mechanics - Vector space of $mathbb{C}^4$ and its basis, the Pauli matrices

How do I write an arbitrary $2\times 2$ matrix as a linear combination of the three Pauli Matrices and the $2\times 2$ unit matrix?

Any example for the same might help ?

Answer

The matrices $\sigma_0\equiv \boldsymbol{1}_2$, $\sigma_x$, $\sigma_y$ and $\sigma_z$ form an orthonormal basis of your vector space w.r.t. the scalar product $$ (X,Y) \equiv \frac{1}{2}\operatorname{tr}(X\cdot Y), $$ where $X$ and $Y$ label any two complex $2\times 2$ matrices. The factor $1/2$ is just for convenience, you may as well normalise your Pauli matrices by dividing them by $2$.

All you want to do now is to decompose an arbitrary element $$ M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ of your vector space into the above basis and figure out the coefficients. As usual, this is done by projecting onto that basis by means of the scalar product $$ M = (\sigma_0,M)\cdot\sigma_0 + (\sigma_x,M)\cdot\sigma_x + (\sigma_y,M)\cdot\sigma_y + (\sigma_z,M)\cdot\sigma_z\ . $$

This has essentially been said in the above comments, particularly in the link posted by Kostya.