Cayley-Hamilton Theorem

Theorem

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

Let $\phi$ be an endomorphism of $M$ such that $\map \phi M \subseteq \mathfrak a M$.

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 $\mathbf N = \sqbrk {a_{i j} }$ be an $n \times n$ matrix with entries in $A$.

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

Let $\map {p_{\mathbf N} } x$ be the determinant $\map \det {x \cdot \mathbf I_n - \mathbf N}$.

Then:

$\map {p_{\mathbf N} } {\mathbf N} = \mathbf 0$

as an $n \times n$ zero matrix.

That is:

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

where the $b_i$ are the coefficients of $\map {p_{\mathbf N} } x$.