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 $A = P^{-1}CP$
The companion matrix to the polynomial $(x^2-1)^2=1-2x^2+x^4$ is $$\pmatrix{0&0&0&-1\cr1&0&0&0\cr0&1&0&2\cr0&0&1&0\cr}$$ which has Jordan form $$\pmatrix{1&1&0&0\cr0&1&0&0\cr0&0&-1&1\cr0&0&0&-1\cr}$$ which has more nonzero entries.