Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?

I guess it has to be its Jordan normal form but I am not sure.

Remarks:

A matrix is sparser than other if it has less nonzero entries.

Two square $n \times n$ matrices $A,C$ are similar if there exists and invertible matrix $P$ such that $A = P^{-1}CP$