Finite spinor transformation I am currently studying  the finite spinor transformations in QFT.
There is equation which i do not fully understand. Rather i don't understand the notation and what it represents:
In the script, we are observing a boost in the $x$-Direction.

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*This is the equation i am having trouble understanding: $\sigma_{\mu\nu}*I^{\mu\nu}=2\sigma_{01}$ how did the book reached this result?

I don't know how to write  a matrix here but i can tell you what $I^{\mu\nu}$ contains. $I^{\mu\nu}$ is a 4x4 matrix. It contains a "1" in position  01 (row 0 column 1 ) and it has a "-1" in position 10 (row 1 column 0 ).the rest are zeros.
And my second question is:


*If $A^{\mu\nu}$ can be written in a matrix form, how is this different then $A^\mu_\nu$,when this one is also written in matrix form?

 A: Based on the definition of $I^{\mu\nu}$ you gave in the comment, the result simply comes by expanding the sum over repeated indices. Since the only elements of $I$ that are different from 0 are $I^{01}$ and $I^{10}$, then $\sigma_{\mu\nu}I^{\mu\nu} = \sigma_{01} - \sigma_{10}$. Now because $\sigma$ is antisymmetric $\sigma_{01} - \sigma_{10}=2\sigma_{01}$. Maybe it's easier if you don't really think of tensors as matrices, but just as collections of elements labelled by indices. In this case $\sigma$ is a collection of matrices, and $I$ is a collection of numbers, so this particular tensor product is a linear combination of matrices.
As for the second question, assume our convention for the Minkowski metric is $\eta_{\mu\nu}=diag(1,-1,-1,-1)$. In this case, when going from $A^{\mu\nu}$ to ${A^\mu}_\nu$ elements whose second index is "space-like" index (i.e. 1, 2 or 3) will get multiplied by -1. You can convince yourself of this by just writing down ${A^\mu}_\nu=A^{\mu\rho}\eta_{\rho\nu}$, fixing the indices $\mu$ and $\nu$ and expanding the sum over repeated indices for several couples of indices. So if you want to visualize these tensors as matrices,
$$A^{\mu\nu}=\begin{pmatrix}
A^{00} & A^{01} & A^{02} & A^{03}\\
A^{10} & A^{11} & A^{12} & A^{13}\\
A^{20} & A^{21} & A^{22} & A^{23}\\
A^{30} & A^{31} & A^{32} & A^{33}\\
\end{pmatrix},\qquad
{A^\mu}_\nu=\begin{pmatrix}
A^{00} & -A^{01} & -A^{02} & -A^{03}\\
A^{10} & -A^{11} & -A^{12} & -A^{13}\\
A^{20} & -A^{21} & -A^{22} & -A^{23}\\
A^{30} & -A^{31} & -A^{32} & -A^{33}\\
\end{pmatrix} 
$$
