I wanted to find explicit elements of inverse transformation of any Lorentz transformation with matrix form: $$L=\begin{pmatrix}L^0_{0}&amp;L^0_{1}&amp;L^0_{2}&amp;L^0_{3}\\L^1_{0}&amp;L^1_{1}&amp;L^1_{2}&amp;L^1_{3}\\L^2_{0}&amp;L^2_{1}&amp;L^2_{2}&amp;L^2_{3}\\L^3_{0}&amp;L^3_{1}&amp;L^3_{2}&amp;L^3_{3}\end{pmatrix}$$ How to derive this elements for this matrix and what with convention?
By definition a matrix representing a Lorentz transformation is orthogonal, so that its inverse is equal to its transpose.   Consider a pure boost in the t-x plane; $$\Lambda_x=\begin{pmatrix} \cosh(\gamma) &amp;&amp; \sinh(\gamma) &amp;&amp; 0 &amp;&amp; 0\\ \sinh(\gamma) &amp;&amp; \cosh(\gamma) &amp;&amp; 0 &amp;&amp; 0\\ 0 &amp;&amp; 0 &amp;&amp; 1 &amp;&amp; 0\\ 0&amp;&amp;0&amp;&amp;0&amp;&amp;1 \end{pmatrix}.$$ $\Lambda_x$ has inverse  $$\Lambda_x^{-1}=\begin{pmatrix} \cosh(\gamma) &amp;&amp; -\sinh(\gamma) &amp;&amp; 0&amp;&amp;0\\ -\sinh(\gamma) &amp;&amp; \cosh(\gamma) &amp;&amp; 0&amp;&amp;0\\ 0 &amp;&amp; 0 &amp;&amp; 1 &amp;&amp;0\\ 0&amp;&amp;0&amp;&amp;0&amp;&amp;1 \end{pmatrix}$$ but tranpose  $$\Lambda_x^T=\begin{pmatrix} \cosh(\gamma) &amp;&amp; \sinh(\gamma) &amp;&amp; 0&amp;&amp;0\\ \sinh(\gamma) &amp;&amp; \cosh(\gamma) &amp;&amp; 0&amp;&amp;0\\ 0 &amp;&amp; 0 &amp;&amp; 1&amp;&amp;0\\ 0&amp;&amp;0&amp;&amp;0&amp;&amp;1 \end{pmatrix}.$$ These are not equal. Where have I gone wrong?
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