Characteristic Polynomial

Let $T$ be a linear operator on a finite-dimensional vector space $V$ over a field $F$, and let $n=\dim(V)$. The characteristic polynomial of $T$ is

where $\det$ is the determinant and $F[t]$ is the polynomial ring in one indeterminate $t$ over $F$. The polynomial $\chi_T(t)$ is monic of degree $n$.

Over a field in which $\chi_T$ splits, the roots of $\chi_T$ are precisely the eigenvalues (with multiplicities). The Cayley-Hamilton theorem says that $T$ satisfies its own characteristic polynomial.

Examples:

• For $A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$, $\chi_A(t)=t^2-(a+d)t+(ad-bc)$.
• If $A$ is diagonal with diagonal entries $\lambda_1,\dots,\lambda_n$, then $\chi_A(t)=\prod_{i=1}^n (t-\lambda_i)$.
• For the nilpotent Jordan block $J=\begin{pmatrix}0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots& &\ddots&\ddots&\vdots\\ 0&\cdots&0&0&1\\ 0&\cdots&\cdots&0&0\end{pmatrix}$, one has $\chi_J(t)=t^n$.