Characteristic polynomial

A characteristic polynomial of a linear operator $T:V\to V$ on an $n$ -dimensional vector space is the polynomial

p_T(t)=\det(tI-A)\in\mathbb{F}[t],

where $A$ is the matrix of $T$ in any basis and $I$ is the $n\times n$ identity matrix. This definition is independent of the chosen basis.

Remarks

The eigenvalues of $T$ in $\mathbb F$ are exactly the roots of $p_T(t)$ in $\mathbb F$ . Over a field extension, the roots are eigenvalues of the operator obtained by extending scalars. The polynomial is central to statements like the Cayley–Hamilton theorem.

Examples

If $A=\operatorname{diag}(d_1,\dots,d_n)$ , then $p_A(t)=\prod_{i=1}^n (t-d_i)$ .
For $A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ , one has $p_A(t)=t^2-(a+d)t+(ad-bc)$ , involving the trace and determinant of $A$ .