Vector space of C4\mathbb{C}^4 and its basis, the Pauli matrices

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

Any example for the same might help ?

Answer

A slow construction would go…

(abcd)=a(1000)+b(0100)+c(0010)+d(0001)


(1000)=12(1001)+12(1001)=1212+12σ3

(0100)= ...


(abcd)=a212+a2σ3+ ... (other combintations of the four matrices)

Attribution
Source : Link , Question Author : chakra , Answer Author : Nikolaj-K

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