I was looking into this general relativity paper from Barone, F., Facchi, P., & Tulczyjew, W. M. (2011). and there's a part where they talk about decomposing a tensor in its parallel and orthogonal components in the following way:
$$ \begin{array}{l} t^{\kappa \lambda}(s)=m(s) \frac{\dot{\xi}^{\kappa} \dot{\xi}^{\lambda}}{\|\dot{\xi}\|^{2}}+m^{\kappa}(s) \frac{\dot{\xi}^{\lambda}}{\|\dot{\xi}\|}+m^{\lambda}(s) \frac{\dot{\xi}^{\kappa}}{\|\dot{\xi}\|}+m^{\kappa \lambda}(s) \\ m(s)=t^{\kappa \lambda} \frac{\dot{\xi}_{\kappa} \dot{\xi}_{\lambda}}{\|\dot{\xi}\|^{2}}, \quad m^{\kappa}(s)=t^{\lambda \mu}\left(\delta_{\lambda}^{\kappa}-\frac{\dot{\xi}_{\lambda} \dot{\xi}^{\kappa}}{\|\dot{\xi}\|^{2}}\right) \frac{\dot{\xi}_{\mu}}{\|\dot{\xi}\|} \\ m^{\kappa \lambda}(s)=t^{\mu \nu}\left(\delta_{\mu}^{\kappa}-\frac{\dot{\xi}_{\mu} \dot{\xi}^{\kappa}}{\|\dot{\xi}\|^{2}}\right)\left(\delta_{v}^{\lambda}-\frac{\dot{\xi}_{\nu} \dot{\xi}^{\lambda}}{\|\dot{\xi}\|^{2}}\right) \end{array} $$
I was wondering how is it possible to express a tensor in this way and where I could find more information about this type of decomposition.