Relationship between matrix rank and beamforming
Typically, when doing any sort of adaptive bamforming, one needs to invert a (square) (covariance) matrix and it needs to be full rank in order to do that. Actually there are work arounds if it isn't full rank and it doesn't always require a literal inversion, like using rank one updates of QR or Cholesky decomposition.  So, that leads into what is a rank one matrix? If $\mathbf{x}$ is a column matrix, the outer product , $$ \mathbf{X} = \mathbf{x} \mathbf{x}^H
$$ is a rank one matrix. A matrix of rank $N$ is, assuming each $\mathbf{x}$ span a linear space, is a sum of $N$ rank one matrices.