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The Lorentz boost transforms 4 vectors from one intertial reference frames to the other. The common form of this transformation is

 c t \rightarrow ct \gamma + \beta \gamma x

 x \rightarrow \gamma x + \beta \gamma c t

 y \rightarrow y

 z \rightarrow z

where \gamma = \left(1-\beta^2\right)^{-1/2},  \beta = v/c ,  v is the relative velocity between the two reference frames and  c is the speed of light. We would like to generalise this transformation to a general velocity vector  \vec{\beta} . To so this, we split the position vector into a component parallel and perpendicular to  \vec{\beta} . The parallel component  x = \vec{\beta} \cdot \vec{r} / \beta transforms in the way described above, and the perpendicular component does not change. Hence

 c t \rightarrow c t \gamma + \gamma \vec{\beta} \cdot \vec{r}

 \vec{r} \rightarrow \gamma \frac{\vec{\beta} \cdot \vec{r}}{\beta^2} \vec{\beta} + \vec{r} - \frac{\vec{\beta} \cdot \vec{r}}{\beta^2} \vec{\beta} +  \gamma \vec{\beta} c t

Hence the transformation matrix can be written in the following form

 \Lambda \left( \vec{\beta} \right) = \begin{pmatrix}
\gamma  & \gamma \vec{\beta}\\ 
\gamma \beta^T & 1 + \left(\gamma-1 \right )\frac{\vec{\beta}^T\vec{\beta}}{\beta^2}
\end{pmatrix}

where  \vec{\beta}^T \vec{\beta} is an outer an outer product (i.e. a tensor) rather than a scalar.