Question

Given any polynomial `p(x)` is there a matrix for which it is the minimum polynomial ?

Answer 1

Yes, the companion matrix works.

Explanation:

Given a monic polynomial:

`p(x) = x^n+a_(n-1) x^(n-1)+a_(n-2) x^(n-2)+...+a_0`

Its companion matrix is an `n xx n` matrix with form:

`C(p) = ((0, 0, 0, ..., 0, -a_0), (1, 0, 0, ..., 0, -a_1), (0, 1, 0, ..., 0, -a_2), (0, 0, 1, ..., 0, -a_3), (vdots, vdots, vdots, ddots, vdots, vdots), (0, 0, 0, ..., 1, -a_(n-1)))`

For example, the companion matrix of `x^5+4x+2` is:

`((0, 0, 0, 0, -2), (1, 0, 0, 0, -4), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0))`

The characteristic polynomial and the minimum polynomial of `C(p)` is `p(x)`.

Remarks

When I first came across the companion matrix I was quite excited: Here is a matrix which generates a field of matrices containing a root of a given polynomial. The rational numbers embed in this field of matrices as rational multiples of the identity matrix.

However, the companion matrix is not identifiable as a particular real or complex number in relation to the rational numbers - it just behaves exactly as a root of the monic polynomial. The same effect can be achieved by considering the ring of polynomials with rational coefficients modulo `p(x)`.