companion matrix

Given any monic polynomial p(x) = x^n + c_(n-1) x^(n-1) + ... + c_1 x + c_0, the companion matrix C(p) is a specific n-by-n matrix built to REALIZE p as an operator. You place 1s on the subdiagonal and write the negated coefficients down the last column (or last row, depending on convention). It is the matrix engineered so that its algebra matches the polynomial exactly.

Its defining feature is striking: both the characteristic polynomial AND the minimal polynomial of C(p) equal p. So for any monic polynomial you choose, there is an operator whose entire eigen-story — eigenvalues, their full multiplicities, the works — is dictated by p, with the minimal and characteristic polynomials coinciding. This happens precisely because C(p) is cyclic: a single vector e_1 generates the whole space under repeated application of C.

The reason char = min here is the cyclic structure. The vectors e_1, C e_1, C^2 e_1, ..., C^(n-1) e_1 are linearly independent (they are essentially the standard basis), so no polynomial of degree below n can annihilate C. Thus the minimal polynomial has degree n, but it always divides the degree-n characteristic polynomial, forcing the two to be equal.

Why it matters: companion matrices prove that every monic polynomial is somebody's characteristic polynomial, they are the building blocks of the rational canonical form (which stacks companion blocks of the invariant factors), and they convert an n-th order linear recurrence or ODE into a first-order system you can attack with matrix methods.