Characteristic polynomial

Definition

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$. The characteristic polynomial of $A$ is the polynomial $$p_A(t) = \det(t I_n - A) \in K[t],$$ where $\det$ is the determinant of a matrix with entries in the polynomial ring $K[t]$.

Remarks

The polynomial $p_A$ is monic of degree $n$. For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $p_A(t) = t^2 - (a + d) t + (ad - bc)$. Some authors use $\det(A - t I_n)$, which equals $(-1)^n p_A(t)$. Over a field, the roots of $p_A$ are the eigenvalues of $A$.

Explanations

L1 human

The number $\lambda$ is an eigenvalue exactly when $A - \lambda I$ is singular, that is, when $\det(\lambda I - A) = 0$. If we replace $\lambda$ by a variable $t$, then we get a polynomial. Its roots are the eigenvalues.