Similarity Classes of Matrices

Let F be any field.

Definition A.1. Two matrices A and B with entries in F are said to be similar if there exists an invertible matrix X such that BX = XA.

Similarity is an equivalence relation on the set of all n×n matrices. The
equivalence classes are called similarity classes. Given a matrix A M_{n}(F),
for every vector x F^{n} and every polynomial f(t) F[t] define fx = f(A)x.
This endows F^{n} with the structure of an F[t]-module, which will be denoted
by M^{A}.

Conversely, given an F[t]-module M, pick any basis of M as an
F-vector space. Let A_{M} 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 A_{M}. Therefore, M determines a similarity class of
matrices.

Proposition A.3. AM^{A} gives rise to a bijection between the
set of similarity classes of matrices and the set of isomorphism classes
of F[t]-modules.

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 M^{A} is a simple
F[t]-module.

Exercise A.6. For any two matrices A and B, let A⊕B denote the
block matrix . A⊕B will be called the direct sum of A and B. Show
that M^{A⊕B} = M^{A} ⊕ M^{B} (a canonical isomorphism of F[t]-modules).

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 M^{A} 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 M^{A} is a semisimple F[t]-module (i.e.,
M^{A} is a direct sum of simple F[t]-modules).

Exercise A.9. For any λ F, show that the matrix is indecomposable, but not semisimple (and hence not simple either).

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 M_{f} = 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 = M_{f}. M is called primary if it is f-primary for some
f.

Exercise A.11. Let f(t) F[t] be an irreducible monic polynomial,
and p(t) F[t] be any monic polynomial. Show that F[t]∕p(t) is f-primary
if and only if p(t) = f(t)^{r} for some r ≥ 0.

Theorem A.12. Let f(t) F[t] be an irreducible monic polynomial,
and A be a square matrix. Then M_{f}^{A}0 if and only if f(t) divides the
characteristic polynomial of A.

Proof. Let χ_{A} denote the characteristic polynomial of A. If f is
an irreducible polynomial that does not divide χ_{A}, then there exist
polynomials r and s such that fr+χ_{A}s = 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, M_{f}^{A} = 0.

Conversely, if M_{f}^{A} = 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 M^{A} 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.

Corollary A.13. Every matrix A M_{n}(F) is similar to a matrix of
the form

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 f_{1},…,f_{r} such that
f_{1}∣∣f_{r} and

Fix an irreducible monic polynomial f(t) F[t]. If M is f-primary, then by
Exercise A.11, each for each i, f_{i} = f^{λi} for some λ_{i} > 0. Therefore,

Corollary A.15 (Structure of a primary module). If M is an
f-primary F[t]-module, then there exists a non-decreasing sequence of
integers λ_{1} ≤≤ λ_{r} such that

Definition A.16 (Partition). A partition is a finite sequence
λ = (λ_{1},,λ_{r}) of positive integers such that λ_{1} ≤ ≤ λ_{r}. Define
∣λ∣ := λ_{1} + + λ_{r}. 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 = (λ_{1},…,λ_{l}), define an F[t]-module

Exercise A.17. Suppose that f and f′ are two irreducible monic
polynomials, λ and λ′ two partitions. Show that the F[t]-modules M_{f,λ}
and M_{f′,λ′} are isomorphic if and only if f = f′ and λ = λ′.

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 M_{φ}
denote the F[t]-module

Theorem A.18 (Similarity classes of matrices). The map ψM_{ψ}
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 n_{ψ} = 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:

Theorem A.19. Suppose that f an irreducible monic polynomial
in F[t] such that f′(t) is not identically zero. Let E denote the field
F[t]∕f(t). Then the rings k[t]∕f(t)^{r} and E[u]∕u^{r} are isomorphic.

Proof. The main step in the proof is a version of Hensel’s Lemma

Lemma A.20 (Hensel). There exists q_{r}(t) F[t] such that q_{r}(t) ≡ t
mod f(t), and f(q_{r}(t)) ≡ 0 mod f(t)^{r}.

Proof. The proof is by induction on r. When r = 1, one may take
q_{1}(t) = t.

Suppose that q_{r-1}(t) F[t] is such that

Given q_{r}(t) as in Hensel’s lemma, the map

Definition A.21 (Companion matrix). Let f(t) = t^{n}-a_{n-1}t^{n-1} --a_{1}t-a_{0}.
Then the companion matrix of f is the n × n matrix:

Theorem A.22 (Block Jordan Canonical Form). Let A M_{n}(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 M_{n}(F) define

Theorem A.23. Let A M_{n}(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 ⊕_{f}A_{f}, where A_{f} is f-primary
(see Corollary A.13). Then Z(A) ⊕_{f}Z(A_{f}). If A is f-primary, E = F[t]∕f(t),
and λ is the partition associated to M^{A} in Corollary A.15, then

Note that the group of units of the centraliser algebra Z(A) will be the
centraliser of A in GL_{n}(F).

Proof. The theorem follows easily from Theorem A.19, using the
fact that End_{F[t]}M^{A}Z(A). □

Definition A.24. A perfect field is either a field of characteristic
zero, or a field of characteristic p > 0 for which the map xx^{p} is
bijective.

Theorem A.25. Suppose that F is a perfect field and f(t) F[t] is a non-constant irreducible polynomial. Then f′(t) does not vanish identically.

Proof. If f′ = 0, then the characteristic of F must be p > 0 and f must be of the form