MATH319 Slides 19 Characteristic polynomial 21 Companion matrix

20 All monic polynomials are characteristic polynomials

Proposition (Characteristic polynomials of companion matrix)

The characteristic polynomial of the companion matrix $A_{c}$ is

\det(\lambda I-A_{c})=\lambda^{n}+a_{n-1}\lambda^{n-1}+\dots+a_{1}\lambda+a_{0}.

Thus any monic complex polynomial arises as the characteristic polynomial of some complex matrix.

Proof. We prove this by induction on $n$ . Let $P_{n}$ be the statement that the above identity holds for $n$ . Then $P_{1}$ is trivially true. Assume that the identity holds for $1,\dots,n-1$ and consider $P_{n}$ . We expand the determinant by the first column, and obtain $\det(\lambda I-A_{c})=$