MATH319 Slides 18 Cofactors 20 All monic polynomials are characteristic polynomials

19 Characteristic polynomial

Definition (Characteristic polynomial)

The characteristic polynomial of a $(n\times n)$ complex matrix $A$ is $\chi_{A}(\lambda)=\det(\lambda I-A)$ , where $I$ is the $(n\times n)$ identity matrix. In MATH220, the lectures used $c_{A}(\lambda)=\det(A-\lambda I)$ . The definition used in the MATH319 lectures is standard in control theory. Also

\chi_{A}(\lambda)=\det(\lambda I-A)=\lambda^{n}-\lambda^{n-1}{\hbox{trace}}(A)% +\dots+(-1)^{n}\det A.

Lemma

If $\det(sI-A)\neq 0$ , then $sI-A$ is invertible and

(sI-A)^{-1}=(\det(sI-A))^{-1}{\hbox{adj}}(sI-A).

This may or may not be an appropriate formula for computing the inverse, depending on the size and shape of the matrices.