If $B$ is a $m \times n$ rank-deficient matrix and $A$ is a $m \times n$ full-rank matrix, is it possible to say something about the smallest eigenvalues of the matrix $(A-B)'(A-B)$? By smallest I mean the $3$ smallest eigenvalues if the nullity of $B$ is $3$ ($3$ zero eigenvalues) and so on.
Remarks: 1) $A$ and $B$ do not necessarily commute 2) $A-B$ can be thought as any other linear combination if you wish 3) Ideally I would like to get an upper-bound on the smallest eigenvalues or something like that.