Wednesday, May 24, 2017

Pauli-decomposition

any complex $2\times2$-matrix $A_{\mu\nu}$ (for instance, the lensing Jacobian) can be decomposed in terms of 3 Pauli-matrices $\sigma^{(n)}_{\mu\nu}$ and the unit matrix $\sigma^{(0)}_{\mu\nu}$,
\begin{equation}
A_{\mu\nu} = \sum_{n=0}^3 a_n\sigma^{(n)}_{\mu\nu}.
\end{equation}
can you show that the coefficients are given by $a_n = (A_{\mu\nu}\sigma^{(n)}_{\nu\mu})/2$ and that the set of matrices is a complete basis?

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