One of the nicest theorems in linear algebra is the one that a matrix satisfies its own characteristic polynomial, the so-called Cayley-Hamilton theorem. What is a good way to prove it? In particu...
I see many proofs for the Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem?
Is there a simpler, more abstract proof of the Cayley-Hamilton theorem ...
The statement of the Cayley-Hamilton Theorem is fairly straight-forward. I now know how to find characteristic polynomials from a given matrix (or at least a matrix with certain properties that I am
The Cayley-Hamilton theorem holds in any commutative ring, and even in any commutative semiring (in a sligthly modified version). I am afraid your approach would not recover these results.
Possibly related: Is there a simpler, more abstract proof of the Cayley-Hamilton theorem for matrices?.
0 I'm currently studying the Cayley-Hamilton theorem for an exam, and I do not quite get the proof presented in the lecture. It was structured as follows: first we'll prove it over $\mathbb {C}$ using the fact that every matrix is trigonalizable over $\mathbb {C}$ and then generalize to an arbitrary field $\mathbb {K}$.
14 There are many, many ways to prove the Cayley-Hamilton theorem, and many of them have something to offer in the way of simplicity, generality, or of providing insight into why such a result could be expected to hold in the first place.
3 A consequence of the Cayley-Hamilton theorem is that $$A^n=aI+bA+cA^2\tag 1$$ for some scalar coefficients $a$, $b$ and $c$.