Characteristic polynomial of A and -A (where A is a companion matrix)

Question

Is there something we can say about the characteristic polynomial of $A$ and $-A$ where

• $A$ is a $n \times n$-matrix;
• $A$ is a companion matrix?

I have found an example where $$A = \begin{pmatrix} 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 1 \\ 0 & -1 & 0 & 2 \\ 0 & 0 & -1 & 0 \end{pmatrix}$$ which is almost a companion matrix but with $-1$ instead of $1$ on the 2nd diagonal.

Then $\chi_A(x)=x^4+2x-x-1$ which I think to be the characteristic polynomial of $-A$ which is then indeed a companion matrix.

Is there something like $\chi_A=-\chi_{-A}$ behind this or is this just a coincidence?

Answer 1

This is not a coincidence. The characteristic polynomial of $A$ is $$p_A(x) = \det(Ix - A) = (-1)^{n} \det(A - Ix) = (-1)^{n} p_{-A}(-x),$$ so the two characteristic polynomials differ only by the signs of

1. The odd degree coefficients (for n even);
2. The even degree coefficients (for n odd).