# Relationship between matrix rank and beamforming

I always encounter the term matrix rank in papers related to beamforming. I am only familiar with the basics of beamforming (delay sum beamformer, basic capon). Can someone explain the significance of matrix rank in beamforming in a layman's term?

• In layman's terms is always a bit hard to grasp, and might be impossible for math things like a rank (because the layman explanation is actually the same as the full definition), but maybe we can find a common base: are you aware what a channel matrix is? – Marcus Müller Aug 23 '17 at 12:45

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.