Many people have told me that Cayley-Hamilton Theorem is their favourite theorem in linear algebra. I also like this theorem a lot (Proof: I am dedicating a blog post about it!)
Let be a linear transformation on a finite-dimensional vector space (over ). In other words, satisfies , and for every scalar . It is clear that -fold composition is also a linear map. So we can ask when satisfies a polynomial identity. In other words, we are interested in finding some coefficients such that where is the identity map, and is the zero map (the map that is identically zero). Since the set of all linear transformations on forms a dimensional vector space, where , it follows that the maps are linearly dependent, so there exists a linear relation between the powers of . As a result, we see that satisfies a polynomial of degree at most .
Cayley-Hamilton Theorem shows that we can do significantly better. Recall the characteristic polynomial . It is clear that is a polynomial of degree . Cayley-Hamilton Theorem states that . I am not going to prove this theorem here. A proof can be found located in wikipedia. I am also going to assume that the reader is familiar with the notion that a linear transformation on a finite-dimensional vector space can be identified with a matrix.
Let me explain why I like this theorem. It is because the theorem is true when the field is replaced by any commutative ring! In other words, if you consider a matrix with entries from some commutative ring , then define (say, using Laplace expansion) as usual. We still have the conclusion . Of course, it doesn’t make sense to talk about vector spaces over arbitrary rings (for this purpose, we use related objects called modules), but the Cayley-Hamilton Theorem still holds in the sense I described. This is closely related with the so-called “determinant trick” (cf. Nakayama’s Lemma) which is a very convenient tool in commutative algebra.
Here is something cool I learned today (from Professor Lior Silberman). We can deduce Cayley-Hamilton Theorem for commutative rings using the result of Cayley-Hamilton Theorem for fields. This could at first seem like a hopeless task, because we are trying to prove something stronger. Here is how the proof goes. Cayley-Hamilton is true for the field of rational numbers . Since is a subring of , Cayley-Hamilton holds for . By this, I just mean that the conclusion holds whenever is a matrix with entries from . Now consider an arbitrary commutative ring . Let be a matrix with entries from , i.e. . We view as an array indeterminates. Then is a matrix with polynomials (each of them being multivariate in variables with coefficients in ). Now, when these indeterminates are replaced by integers, we get that vanishes (precisely because Cayley-Hamilton is true for ). Thus, vanishes on all of . But each entry of is a polynomial of degree , and no non-zero multivariate polynomial with coefficients in can vanish on all of . It follows that each entry of is the zero polynomial. Consequently, .
It is natural to wonder why plays an important role in the proof above. One of the reasons for this phenomena is that is the initial object in the category of rings.