Is it possible to do Smith Normal form where the matrix has "variable" entries?
For instance consider the matrix: $$\begin{pmatrix} -b & 0 & 0 &-d\\ a & -c & 0 & 0\\ 0 & b& -d & 0\\ 0 & 0 & c &a \end{pmatrix}$$ where $a,b,c,d$ are integers.
Is it possible to use the usual Smith Normal Algorithm to make the matrix into Smith Normal form, with explicit formulae in terms of $a,b,c,d$?
Looking at the algorithm, https://en.wikipedia.org/wiki/Smith_normal_form, it sure looks hard especially since in step II we need the Bezout property which is "existential" in the sense that there is no explicit formula to compute which linear combinations equal to the gcd.
This seems to be unlike Gaussian elimination where the algorithm works equally well for "variables".
Thanks for any help.