TITLE: What does matrix rank $k$ to precision $\epsilon$ mean?
QUESTION [5 upvotes]: I read this sentence.

Suppose that the matrix $A_{ij}$ of dimension $n_i \times n_j$ has rank $k$ to precision $\epsilon$, then there exists a factorization of $A_{ij}$ of the form:
  $A_{ij} = L_i S_{ij} R_j + \text{O}(\epsilon)$.

I wonder what does matrix rank $k$ to precision $\epsilon$ mean? 
Thank you.

REPLY [9 votes]: The rank of a matrix is the number of its nonzero singular values.
Rank to precision $\epsilon$ means that in computing the rank of the matrix, we consider every singular value of the matrix that is less than $\epsilon$ as zero.
This is also known as "numerical rank": the number of singular values greater than $\epsilon$.