I have a (probably singular) square matrix whose entries are given by some complicated formula, involving, in particular, a converging infinite sum. I can determine each entry with arbitrary precision, but I cannot write the exact matrix in a "simple" form.
Is it possible to determine numerically what rank it has?
I think that this is not possible: if I compute the entries up to a certain precision, there will still be an invertible matrix closer to my matrix than my approximation, because they are dense, therefore I cannot conclude.
I have thought about finding a lower bound for the non-zero singular values of my matrix. I would then have shown that the computed values below this bound are, in fact, zero. And note that the rank equals the number of non-zero singular values.
But how would one go about looking for a lower bound for the singular values of a matrix that can only be known up to (arbitrary) precision?