I can't remember the proper terminology I heard long ago for the relationship between matrices like these two.
A and B are rank-deficient square matrices that cover independent subspaces of a transform (and I'm not even sure I phrased that correctly). Their sum is full rank, and its inverse is the sum of the pseudo-inverses.
$|A|=|B|=0$
$AB=BA=0$
$\left(A+B\right)^{-1}=A^{\dagger}+B^{\dagger}$
Thanks!