Characteristic polynomial

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

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.

The eigenvalues of $T$ are exactly the roots of $p_T(t)$ (in any field extension where the polynomial splits). 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 $A$.