The Stacks project

Lemma 10.16.1. Let $R$ be a ring. Let $A = (a_{ij})$ be an $n \times n$ matrix with coefficients in $R$. Let $P(x) \in R[x]$ be the characteristic polynomial of $A$ (defined as $\det (x\text{id}_{n \times n} - A)$). Then $P(A) = 0$ in $\text{Mat}(n \times n, R)$.

Proof. We reduce the question to the well-known Cayley-Hamilton theorem from linear algebra in several steps:

  1. If $\phi :S \rightarrow R$ is a ring morphism and $b_{ij}$ are inverse images of the $a_{ij}$ under this map, then it suffices to show the statement for $S$ and $(b_{ij})$ since $\phi $ is a ring morphism.

  2. If $\psi :R \hookrightarrow S$ is an injective ring morphism, it clearly suffices to show the result for $S$ and the $a_{ij}$ considered as elements of $S$.

  3. Thus we may first reduce to the case $R = \mathbf{Z}[X_{ij}]$, $a_{ij} = X_{ij}$ of a polynomial ring and then further to the case $R = \mathbf{Q}(X_{ij})$ where we may finally apply Cayley-Hamilton.

$\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00DX. Beware of the difference between the letter 'O' and the digit '0'.