Orthogonal projection matrix

A complex matrix $P$ represents an orthogonal projection exactly whenIts image and kernel are orthogonal complements.

For a nonnull vector $k$ in a space with a nondegenerate bilinear form, $P_{\mu \nu }={\eta }_{\mu \nu }-k_{\mu }{k}_{\nu }/k^2$ projects onto the subspace transverse to $k$. It obeys $P_{\mu \nu }{k}^{\nu }=0$ and $P^2=P$.