The idea of this post is for people to post different proofs of the Cayley-Hamilton Theorem. You can either try to post your own proof or give a reference. If you usse a reference, please give some ideas of what the proof looks like ;). I really encourage you to upvote this post in order to have a good internet reference for these proofs.
The Cayley-Hamilton Theorem states that if $A$ is a square matrix with coefficients in any unital, commutative ring $R$, and $p$ is its characteristic polynomial, given by $p(t) = \det(t I-A)$ then $p(A)=0$.
I would also like to fit these solutions into categories. Here are the proofs I am aware of, in their respective categories:
Proofs in the case $R=\mathbb C$ using topology (for example, using that the theorem is true for diagonalisable matrices and showing that the diagonalisable matrices are dense to infer the general theorem, as here) or complex analysis (a proof using Cauchy's formula can be found here). Note that if the theorem is valid over $\mathbb C$, it is valid over any ring so we are not losing any generality.
Proofs in the case $R$ is an algebraically closed field using the Jordan form of a matrix (I think this is the most standard proof and there are many posts about this proof, such as this).
Proofs in the case $R$ is any field, using theory of endomorphisms and the decomposition of a vector space into cyclic subspaces (one such proof can be found in this answer).
Direct proofs of the matrix identity $A^n + c_{n-1}A^{n-1} + \ldots + c_0=0$ in any ring:
- By direct computation (a 4 page long proof can be found in 3-4/2014 issue of Gazeta Matematica, Seria A, pages 32-36; available online here)
- By induction in $n$ (https://digital.library.unt.edu/ark:/67531/metadc407866/m1/1/)
- Using some ingenuous manipulations using the matrix $\text{adj}(tI-A)$, as in the proofs presented in the Wikipedia page for the Cayley-Hamilton theorem.
- Ring theoretic proofs like the one given in this comment. The idea is to show that if $a_{ij}$ are the entries of $A$, the characteristic polynomial $p$ as a polynomial with coefficients in the quotient field of $\mathbb Z [a_{ij}]_{i,j}$ is separable and use that to reduce to the diagonal case.
EDIT: For the sake of clarity, since this post has raised so much concern, let me clarify that an answer to this post should be a proof of the Cayley-Hamilton Theorem (which you can reformulate in terms of multilinear algebra or what you like the most) different from the ones listed above. The categories should help you decide if your proof could be one of the list and check it using the link
