The following matrix A is,(adsbygoogle = window.adsbygoogle || []).push({});

\begin{equation}

A=

\begin{bmatrix}

a+b-\sigma\cdot p & -x_1 \\

x_2 & a-b-\sigma\cdot p

\end{bmatrix}

\end{equation}

The inversion of matrix A is,

\begin{equation}

A^{-1}=

\frac{\begin{bmatrix}

a-b-\sigma\cdot p & x_1 \\

-x_2 & a+b-\sigma\cdot p

\end{bmatrix}}{a^2-b^2+p^2-2\sigma\cdot p-x_1x_2}

\end{equation}

The textbook shows the formula in different form,

\begin{equation}

A^{-1}=

\frac{(1/2)\begin{bmatrix}

(\sigma_o+\sigma\cdot \hat{p})(a-b+p) & (\sigma_o+\sigma\cdot \hat{p}) x_1 \\

-x_2(\sigma_o+\sigma\cdot \hat{p}) & \sigma_y(\sigma_o+\sigma\cdot \hat{p})\sigma_y(a+b-p)

\end{bmatrix}}{a^2-b^2-p^2+2bp-x_1x_2}

\end{equation}

and p hat is p/b and sigma's are 2x2 pauli matrices. The sigmas inside brackets are projectors. How this projector was derived? How projector is used to get the inversion of matrix A that looks different from conventional method used to calculate A^(-1)?

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# A How spin projector got included in inverse of Matrix?

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