Cayley-Hamilton Theorem

Theorem

Let $\mathfrak a$ be an ideal of $A$.

Let $\phi$ be an endomorphism of $M$ such that $\phi \left({M}\right) \subseteq \mathfrak a M$.

Then $\phi$ satisfies an equation of the form:

$\phi^n + a_{n-1} \phi^{n-1} + \cdots + a_1 \phi + a_0 = 0$

with the $a_i \in \mathfrak a$.

Let $\mathbf N = \left({a_{ij} }\right)$ be an $n \times n$ matrix with entries in $A$.

Let $\mathbf I_n$ denote the $n \times n$ unit matrix.

Let $p_N \left({x}\right)$ be the determinant $\det \left({x \cdot \mathbf I_n - \mathbf N}\right)$.

Then:

$p_N \left({N}\right) = \mathbf 0$

as an $n \times n$ zero matrix.

That is:

$ N^n + b_{n-1} N^{n-1} + \cdots + b_1 N + b_0 = \mathbf 0$

where the $b_i$ are the coefficients of $p_N \left({x}\right)$.