Let F be any field.
Similarity is an equivalence relation on the set of all n×n matrices. The equivalence classes are called similarity classes. Given a matrix A Mn(F), for every vector x Fn and every polynomial f(t) F[t] define fx = f(A)x. This endows Fn with the structure of an F[t]-module, which will be denoted by MA.
Conversely, given an F[t]-module M, pick any basis of M as an F-vector space. Let AM be the matrix by which t acts on M with respect to this basis. A different basis of M would give rise to a matrix similar to AM. Therefore, M determines a similarity class of matrices.
Definition A.4 (Simple matrix). Recall that an F[t]-module is called simple if there is no non-trivial proper subspace of M which is preserved by F[t]. A matrix A is said to be simple if MA is a simple F[t]-module.
Definition A.7 (Indecomposable matrix). A matrix is said to be indecomposable if it is not similar to a matrix of the form A ⊕ B, where A and B are two strictly smaller matrices. Equivalently, A is indecomposable if MA is indecomposable as an F[t]-module.
Definition A.8 (Semisimple matrix). A matrix is said to be semisimple if it is similar to a direct sum of simple matrices. Equivalently, A is semisimple if MA is a semisimple F[t]-module (i.e., MA is a direct sum of simple F[t]-modules).
Let f(t) be any irreducible monic polynomial in F[t]. Given an F[t]-module M, its f-primary part is the submodule
Theorem A.10 (Primary decomposition). [Jac84, Theorem 3.11] Let M be an F[t]-module which is also a finite dimensional F-vector space. Then Mf = 0 for all but finitely many irreducible monic polynomials f(t) F[t].
Let f F[t] be an irreducible monic polynomial. An F[t]-module M is called f-primary if M = Mf. M is called primary if it is f-primary for some f.
Proof. Let χA denote the characteristic polynomial of A. If f is an irreducible polynomial that does not divide χA, then there exist polynomials r and s such that fr+χAs = 1. Evaluating at A and applying the Cayley-Hamilton theorem shows that f(A)r(A) = I. It follows that f(A) is non-singular. Hence f(A)k is also non-singular for every positive integer k. Therefore, MfA = 0.
Conversely, if MfA = 0, then f(A)k is non-singular for every k N. In particular, f(A) is non-singular. Let E be a splitting field of f. Suppose that
If MA is f-primary then the matrix A is called an f-primary matrix. It follows that a matrix is primary if and only if its characteristic polynomial has a unique irreducible factor.
Thus, the study of similarity classes of matrices is reduced to the study of similarity classes of primary matrices.
Theorem A.14 (Structure theorem). [Jac84, Section 3.8] For every F[t]-module M, there exist non-constant monic polynomials f1,…,fr such that f1∣∣fr and
Fix an irreducible monic polynomial f(t) F[t]. If M is f-primary, then by Exercise A.11, each for each i, fi = fλi for some λi > 0. Therefore,
Definition A.16 (Partition). A partition is a finite sequence λ = (λ1,,λr) of positive integers such that λ1 ≤ ≤ λr. Define ∣λ∣ := λ1 + + λr. One says that λ is a partition of ∣λ∣. The length of λ is the non-negative integer r (there is an ‘empty partition’ of length 0 denoted ∅, with ∣∅∣ = 0). Let Λ denote the set of all partitions.
Given a partition λ = (λ1,…,λl), define an F[t]-module
Let S denote the set of all irreducible monic polynomials in F[t]. Given a function ψ : S → Λ such that ψ(f) = ∅ for all but finitely many f S, let Mφ denote the F[t]-module
Theorem A.18 (Similarity classes of matrices). The map ψMψ is a bijective correspondence between the set of all functions S → Λ with the property that ψ(f) = ∅ for all but finitely many f S and nψ = n and the set of isomorphism classes of n-dimensional F[t]-modules (and hence the set of similarity classes of n × n matrices).
There is a version of the Jordan canonical form for matrices for which the irreducible factors of the characteristic polynomial have derivatives which are not identically zero.
In order to obtain this form, we need the following result:
Proof. The main step in the proof is a version of Hensel’s Lemma
Proof. The proof is by induction on r. When r = 1, one may take q1(t) = t.
Suppose that qr-1(t) F[t] is such that
Given qr(t) as in Hensel’s lemma, the map
Theorem A.22 (Block Jordan Canonical Form). Let A Mn(F) be such that for every irreducible factor f of the characteristic polynomial of A, f′ is not identically zero. Then A can be written as a block diagonal matrix with blocks of the form
Proof. By Exercise A.6 and Theorem A.10 one may assume that A is f-primary, for some irreducible monic polynomial f. Let E = F[v]∕f(v). By Corollary A.15 and Theorem A.15, there exists a partition λ such that
The hypothesis on A in Theorem A.22 always holds when F is a perfect field, as we shall see in Section A.6. By Corollary B.8 every finite field is perfect. Therefore, every matrix over a finite field has a Jordan canonical form.
For any A Mn(F) define
Theorem A.23. Let A Mn(F) be a matrix such that for each irreducible factor f of the characteristic polynomial of A, f′ is not identically zero. Suppose that A is similar to ⊕fAf, where Af is f-primary (see Corollary A.13). Then Z(A) ⊕fZ(Af). If A is f-primary, E = F[t]∕f(t), and λ is the partition associated to MA in Corollary A.15, then
Note that the group of units of the centraliser algebra Z(A) will be the centraliser of A in GLn(F).
Proof. The theorem follows easily from Theorem A.19, using the fact that EndF[t]MAZ(A). □
Proof. If f′ = 0, then the characteristic of F must be p > 0 and f must be of the form