# How to find generators of a transformation?

We have a 4-vector $V^\mu =(V,0,0,V)$ with sametime and $z$ components. then we are asked to find the transformations that are not pure rotations and leave $V$ invariant. Sone one such matrix is the rotation matrix as follows:$$\Lambda^\mu _\nu=\begin{pmatrix}1&0&0&0\\0&cos\theta &sin\theta&0\\0&-sin\theta &cos \theta &0\\0&0&0&1 \end{pmatrix}$$ The other matrix is $$K_3 =\begin{pmatrix}0&0&0&1\\0&0 &0&0\\0&0 &0 &0\\1&0&0&0 \end{pmatrix}$$ and $K_3 +J_3$ where $J_3$ is just the rotation matrix in $xy$ plane with an angle of 90 degree.

How do I go about finding the three generators of my two transformations and a rotation? Thank you.

• – user108787 Sep 27 '16 at 20:14