presentation matrix xI - A

When you turn V into an F[x]-module using an operator A, you need a way to encode that module concretely so you can compute with it. The presentation matrix xI - A is exactly that encoding: a square matrix over the polynomial ring F[x] that tells you all the relations among a chosen set of generators of the module.

Here is the logic. Take the standard basis e1, ..., en of V as generators. The relations come from how x acts: x . e_j = A e_j = sum_i A_ij e_i, which rearranges to (x . e_j) - sum_i A_ij e_i = 0. Packaging all n of these relations as columns gives precisely the matrix xI - A. So V, as an F[x]-module, is the cokernel of the map F[x]^n -> F[x]^n given by xI - A.

Why this is the workhorse: once V is presented by xI - A, the structure theorem becomes a computation. You diagonalize xI - A by invertible row and column operations over F[x] — that is the Smith normal form — and its nonunit diagonal entries are the invariant factors. The presentation matrix is the bridge from operator to module to canonical form.

One thing to keep clean. The presentation matrix is a matrix of polynomials, not a numerical matrix; its determinant is the characteristic polynomial, det(xI - A). Do not confuse the row/column operations allowed here (invertible over F[x], no dividing by nonunits) with ordinary numerical Gaussian elimination over a field — the polynomial constraint is exactly what makes the invariant factors meaningful.