Matrix Rank

Rank of Matrix Products

Full-rank Matrices

Rank Estimates

Rank 1 Matrices

If u ∈ 𝔽m×1 and v ∈ 𝔽n×1v are both nonzero column vectors, then the outer product matrix u vT always has matrix rank 1. Indeed, the columns of the outer product are all proportional to the first column. Thus they are all linearly dependent on that one column, hence the matrix is of rank one.    

 

  1. Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.