551, Fall '00      B2a.    The module of a linear transformation      p. 1

Math 551 – Algebra – Fall 2000

Richard Lyons Rutgers University New Brunswick, New Jersey, USA

B. Abelian Groups and Modules over Principal Ideal Domains

2. Linear transformations; canonical forms.

2a. The module of a linear transformation

We shall be concerned here with a linear transformation T : V → V, where V is a finitedimensional vector space over a field F. For such a setup, the powers T^{n} (n ≥ 0) i.e., the iterated composites of T with itself, are defined, and they too are linear transformations from V to V. Considering T^{0} to be the identity transformation id_{V}, we can then make V into a module over the polynomial ring F [X] in one indeterminate X. Namely given a polynomial f ∈ F [X], say f(X) = a_{n}X^{n} + ⋯ + a_{1}X + a_{0}, and for any v ∈ V, define

fv = a_{n}T^{n}v + ⋯ + a_{1}T v + a_{0}v.

It is straightforward to check that the module axioms are satisfied. We shall call this module V_{T}.

Because F is a field, the ring F [X] is a Euclidean domain (relative to the degree function), so is a PID. Moreover:

Lemma. In this situation, V is a finitely generated torsion module for F [X].

Proof. A basis of V (as vector space) is finite by assumption and certainly generates V as an F [X]-module. So V is finitely generated. To see that it is a torsion module, either a) use the fundamental theorem to deduce that otherwise, it would have a free rank 1 submodule V_{0} ≅ F [X], leading to the contradiction dim V ≥ dim V_{0} = ∞; or b), set n = dim V and argue that given any v ∈ V, the vectors v, T v, …, T^{n}v form a linearly independent set, and then a dependence relation takes the form pv = 0 for some nonzero polynomial p ∈ F [X].

The fundamental theorem will therefore give a lot of information. In order to decode it, we first consider exactly what information is encoded in the module structure of V. This is the same as asking what it means for two such modules to be isomorphic.

p. 2    B2.    Linear transformations; canonical forms.      551, Fall '00

Lemma. Let T : V → V and U : W → W be linear transformations, where V and W are finite-dimensional vector spaces over the same field F. Then the following conditions are equivalent:

a) V_{T} ≅ W_{U} as F [X] − modules;

b) There exists an isomorphism ϕ : V → W of vector spaces such that the diagram

\begin{aligned}T : & V & → & V \\ ↓ & ↓ \\ U : & W & → & W \end{aligned}

commutes;

c) There exist ordered bases B of V and C of W such that the matrices [T]_{B}^{B} and [U]_{C}^{C} coincide.

Proof. If a) holds, then an isomorphism ϕ : V_{T} → W_{U} is in particular a bijective linear transformation and also satisfies ϕ(Xv) = Xϕ(v) for all v ∈ V. This means ϕ(T v) = Uϕ(v) so b) holds. If b) holds, then c) holds for an arbitrary basis B of V and its image C = ϕ(B), which is a basis of W. If c) holds, and B = {b_{1}, …, b_{n}} and C = {c_{1}, …, c_{n}}, then the mapping ϕ(∑ α_{i}b_{i}) = ∑ α_{i}c_{i} satisfies b). Finally if b) holds, then ϕ(T v) = Uϕ(v) for all v ∈ V, whence ϕ(T^{n}v) = U^{n}ϕ(v) follows by an inductive argument, and then since ϕ, T and U are all linear transformations, ϕ(fv) = fϕ(v) for all f ∈ F [X], and ϕ itself is the required isomorphism.

Two square matrices A, A′ of the same size with coefficients in F are called similar (over F) if and only if there is an invertible D of the same size with coefficients in F such that

DAD^{−1} = A′ or equivalently—as D is invertible—DA = A′D.

Given an ordered basis B of V, the invertible matrices of size dim V are just the matrices D = [id_{V}]_{B′}^{B} as B′ ranges over the set of all ordered bases of V, and so if A = [T]_{B}^{B}, then DAD^{−1} = [T]_{B′}^{B′}. Therefore

Lemma. The conditions of the above lemma are also equivalent to:

c’) For some bases B of V and C of W, the matrices [T]_{B}^{B} and [U]_{C}^{C} are similar;

c”) For any bases B of V and C of W, the matrices [T]_{B}^{B} and [U]_{C}^{C} are similar; The isomorphism type of V_{T} determines and is determined by the equivalence class of [T]_{B}^{B} under similarity.

We can therefore interpret the application of the fundamental theorem as a statement about similarity classes of matrices.

A submodule W ⊆ V_{T} is simply a subspace such that T (W) ⊆ W, since this condition immediately implies that fW ⊆ W for all f ∈ F [X]. Such a submodule W exists and has dimension m if and only if T, with respect to some basis, has matrix in upper-triangular “block” form

[ \begin{matrix}X & Y \\ 0 & Z \end{matrix} ]

551, Fall '00             B2b.    Rational canonical form             p. 3

where X is m × m and Z is (n − m) × (n − m). For if T is similar to such a matrix, then that matrix equals [T]_{B}^{B} for some ordered basis B, and the zeroes in the lower left imply that the first m elements of B span a subspace W such that T (W) ⊆ W. Conversely, if W exists, then an ordered basis of W followed by further vectors constitutes an ordered basis B of V for which [T]_{B}^{B} is in upper-triangular block form.

Similarly, to say that V_{T} has a direct sum decomposition V_{T} = V_{1} ⊕ V_{2} as F [X]-modules, where V_{1} and V_{2} have dimension m and n − m, respectively, is equivalent to saying that for some B, [T]_{B}^{B} is in block-diagonal form

[ \begin{matrix}X_{1} & 0 \\ 0 & X_{2} \end{matrix} ] .

If the decomposition exists, one can take B to consist of a basis of V_{1} followed by a basis of V_{2}; and conversely, if the matrix is in this form, V_{1} can be recovered as the span of the first m elements in the ordered basis B, and V_{2} as the span of the last n − m elements.

Let R = F [X]. What does it mean to say that V_{T} ≅ R/fR as R-modules, where 0 ≠ f ∈ R? For simplicity let us assume that f is monic, i.e. has leading coefficient 1. (There is no loss of generality in this assumption since f has a (unique) associate f^{∗} which is monic, and fR = f^{∗}R.) Write f(X) = X^{n} + a_{n−1}X^{n−1} + ⋯ + a_{1}X + a_{0}. Then V_{T} ≅ R/fR if and only if for some basis B,

[T]_{B}^{B} = [ \begin{matrix}0 & 0 & 0 & 0 & ⋯ & 0 & −a_{0} \\ 1 & 0 & 0 & 0 & ⋯ & 0 & −a_{1} \\ 0 & 1 & 0 & 0 & ⋯ & 0 & −a_{2} \\ 0 & 0 & 1 & 0 & ⋯ & 0 & −a_{3} \\ & & & & ⋯ \\ 0 & 0 & 0 & 0 & ⋯ & 1 & −a_{n−1} \end{matrix} ] .b

The matrix on the right is called the companion matrix of the monic polynomial f, written C(f). Thus C(f) is a certain n × n matrix, where n = deg f.

To see this, notice that in \overline{R} = R/fR, every coset \overline{g} has a unique representative g which is a polynomial of degree strictly less than n = deg f, by the division algorithm. Let us write \overline{h} = h + fR. Then the images \overline{1}, \overline{X}, \overline{X^{2}}, …, \overline{X^{n−1}} form a basis of \overline{R} = R/fR, corresponding to a basis B of V. Multiplication by X transforms this to the basis \overline{X}, \overline{X^{2}}, …, \overline{X^{n}}. Therefore the matrix of T with respect to B has as its columns the coordinates of \overline{X}, \overline{X^{2}}, …, \overline{X^{n}} with respect to the basis \overline{1}, \overline{X}, \overline{X^{2}}, …, \overline{X^{n−1}}. Since \overline{f} = 0 we have f(\overline{X}) = 0, so \overline{X^{n}} = −a_{0} − a_{1}\overline{X} − ⋯, and we obtain the companion matrix.

2b. Rational canonical form

With this interpretation, we immediately get “rational canonical form” theorems. For example, from version II we get:

Theorem. Let V be a finite-dimensional vector space and T : V → V a linear transformation. Then there exists a unique r ≥ 0 and unique monic polynomials f_{1}, …, f_{r} such that f_{1} | f_{2} | ⋯ | f_{r} and [T]_{B}^{B} is the block diagonal matrix

[ \begin{matrix}C(f_{1}) & 0 & ⋯ & 0 \\ 0 & C(f_{2}) & ⋯ & 0 \\ & & ⋯ \\ 0 & 0 & ⋯ & C(f_{r}) \end{matrix} ] .

and its matrix version:

Theorem. Let F be a field and A a square matrix over F. Then there exist unique r and monic polynomials f_{1}, …, f_{r} ∈ F [X] such that f_{1} | f_{2} | ⋯ | f_{r} and A is similar to the block-diagonal matrix whose blocks are the companion matrices C(f_{1}), …, C(f_{r}).

This is obtained by taking V = F^{n} to be the space of column vectors, and T : V → V to be defined by T (v) = Av, and applying the previous theorem.

The polynomials f_{1}, …, f_{r} are called the invariant factors of T or A, as the case may be.

Similarly, there is an “elementary divisor” version, corresponding to version III of the fundamental theorem, giving a canonical form for the similarity class of any square matrix over F as a block-diagonal matrix with the blocks being companion matrices of powers of monic irreducible polynomials (unique up to the order of blocks); these prime powers are the “elementary divisors” of the transformation or matrix.

Theorem. Let A be a square matrix over a field F. Then there exist irreducible polynomials p_{1}, …, p_{r} and positive integers n_{1}, …, n_{r} such that A is similar to the block-diagonal matrix whose blocks are C(p_{1}^{n_{1}}), …, C(p_{r}^{n_{r}}). The prime powers p_{1}^{n_{1}}, …, p_{r}^{n_{r}} are uniquely determined up to order.

Example. The invariant factor version implies that every 2×2 real matrix is similar (over R) to exactly one of the following:

[ \begin{matrix}α & 0 \\ 0 & α \end{matrix} ] , α ∈ R or [ \begin{matrix}0 & γ \\ 1 & β \end{matrix} ] , β, γ ∈ R

the first having invariant factors X − α, X − α and the second having the single invariant factor X^{2} − βX − γ. The elementary divisor version implies that every real matrix is similar (over R) to exactly one of the following:

[ \begin{matrix}α & 0 \\ 0 & β \end{matrix} ] , α, β ∈ R or [ \begin{matrix}0 & γ \\ 1 & β \end{matrix} ] , β, γ ∈ R, β^{2} + 4γ ≤ 0.

This is because the monic irreducible polynomials over R are the polynomials X−α, α ∈ R and the polynomials X^{2} +βX +γ with β^{2} < 4γ. (Notice that the companion matrix of this last polynomials has entries −β and −γ in the last column.) Now the elementary divisors of a 2 × 2 matrix either are a) two irreducible polynomials of degree 1; b) an irreducible

551, Fall '00              B2c.    Jordan canonical form              p. 5

polynomial of degree 2; or c) the square of an irreducible polynomial of degree 1. It is the third possibility which is covered above by the case β^{2} + 4γ = 0.

The rational forms for 2 × 2 matrices over C are similar, except that the quadratic irreducible possibility does not exist.

Those of the above forms with nonzero determinant therefore are a set of representatives for the conjugacy classes of the group GL_{2}(R).

Caution: the term “rational canonical form” is not universally interpreted this way.

2c. Jordan canonical form

Definition. The field F is algebraically closed if and only for every nonconstant polynomial f ∈ F [X] there exists a zero, i.e., an element z ∈ F such that f(z) = 0.

This immediately implies that X − z divides f. (Write f(X) = q(X)(X − z) + y, with y ∈ F. Then 0 = f(z) = y.) Consequently over an algebraically closed field, the monic irreducible elements of F [X] are just the linear polynomials X − α, α ∈ F. Conversely, if these are all the irreducibles, then the fact that F [X] is a UFD (uniqueness doesn’t matter) implies that every polynomial has a linear factor, hence a zero, so F is algebraically closed.

The most famous example of an algebraically closed field is C. But in fact every field can be embedded in an algebraically closed field (proof next semester).

To analyze linear transformations and matrices over an algebraically closed field, there is a more useful canonical form than rational canonical form. It uses the elementary divisor version of the fundamental theorem, and “Jordan blocks” instead of companion matrices. Every monic prime power in F [X], in this case, is of the form f(X) = (X −α)^{n}. Instead of the basis \overline{1}, \overline{X}, …, \overline{X^{n−1}} of R/fR which we used for companion matrices, we can use the basis \overline{1}, \overline{X} − α, (\overline{X} − α)^{2}, …, (\overline{X} − α)^{n−1}. We have \overline{X}(\overline{X} − α)^{i} = (α + \overline{X} − α)(\overline{X} − α)^{i} = α(\overline{X} − α)^{i} + (\overline{X} − α)^{i+1}. As (\overline{X} − α)^{n} = 0 we get the lower triangular matrix

J(α, n) = [ \begin{matrix}α & 0 & 0 & ⋯ & 0 & 0 \\ 1 & α & 0 & ⋯ & 0 & 0 \\ 0 & 1 & α & ⋯ & 0 & 0 \\ & & & ⋯ \\ 0 & 0 & 0 & ⋯ & 1 & α \end{matrix} ] ,

all diagonal entries being α, all sub-diagonal entries being 1, and all other entries being 0. Version III of the fundamental theorem then implies (we state only the matrix version):

A matrix will be said to be in Jordan canonical form (over F) if and only if it is in block-diagonal form

[ \begin{matrix}J(α_{1}, n_{1}) & 0 & ⋯ & 0 \\ 0 & J(α_{2}, n_{2}) & ⋯ & 0 \\ & & ⋯ \\ 0 & 0 & ⋯ & J(α_{r}, n_{r}) \end{matrix} ]

for some α_{i} ∈ F and some positive integers n_{1}, …, n_{r}.

Version III of the fundamental theorem then implies:

Theorem. Let F be an algebraically closed field. Then any square matrix over F is similar to a matrix in Jordan canonical form. This canonical form is unique except for the order in which the diagonal blocks appear.

For the above canonical forms to be useful it is important that they can be calculated, or at least partial information about them can be calculated. The important invariants μ_{A} and χ_{A} (minimal and characteristic polynomials) will be discussed below. For Jordan canonical form, there is a nice characterization of the number of blocks J(α, n) for any α ∈ F and any positive integer n, although it requires fairly detailed information about the matrix in question. We define the nullspace N of an n × n matrix A to be the space of all vectors v such that Av = 0, and the nullity ν(A) to be dim N(A). If A = [T]_{B}^{B} (or even [T]_{B′}^{B}!) then ν(A) = dim ker(T). Therefore if A and A′ are similar, then ν(A) = ν(A′).

Exercise. If A is in block diagonal form,

A = [ \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} ] ,

then ν(A) = ν(A_{1}) + ν(A_{2}).

Proposition. Let F be an algebraically closed field and A a square matrix (or T : V → V a linear transformation). Let α ∈ F and let n be a positive integer. Then the number of Jordan blocks J(α, m) such that m > n in the Jordan canonical form of A (or T) is

ν((A − αI)^{n+1}) − ν((A − αI)^{n}), or dim ker((T − α)^{n+1}) − dim ker((T − α)^{n}).

Here we write α for the linear transformation v ↦ αv of V.

Once we know how many blocks J(α, m) there are satisfying m > n, and we know this for all n, subtraction gives us the number of J(α, n) blocks.

Proof. If A and A′ are similar, then so are A−αI and A′ −αI and hence so are (A−αI)^{n} and (A′ − αI)^{n} for every n. Hence the nullities of these two matrices coincide. The upshot of this remark is that in proving our result, we are free to replace A by any matrix similar to it, and so we may assume that A is already in Jordan canonical form. Using the exercise above, we reduce to the case of a single Jordan block A = J(α, n). But then A − β is invertible for all β ≠ α, so the nullities ν((A − β)^{n}) are all 0 and our formula predicts no Jordan blocks J(β, m), which is correct. For α itself, if A = J(α, m), then A−αI = J(0, m), and as one takes the powers of J(0, m), the 1’s slide down one diagonal at a time, until J(0, m)^{m} = 0. Therefore ν(J(0, m)^{i}) = i for 0 ≤ i ≤ m, and ν(J(0, m)^{i}) = m for i > m. Hence

ν(J(0, m)^{n+1}) − ν(J(0, m)^{n}) = \{ \begin{matrix}1 & if 0 ≤ n < m \\ 0 & if n ≥ m \end{matrix} ,

551, Fall '00                B2d.    Minimal polynomial                p. 7

so our formula counts blocks as claimed.

2d. Minimal polynomial

Two of the most important of the invariants of a linear transformation or square matrix (over a field F) are its minimal polynomial and characteristic polynomial, both elements of F [X].

The minimal polynomial is defined as follows. Let V be a finite dimensional vector space over the field F. The space A = Hom_{F}(V, V), in addition to being a vector space over F, carries a (noncommutative) ring structure, with multiplication given by composition of functions. Such a ring/vector space A is called an F -algebra if scalar multiplication and multiplication obey the axiom:

α(xy) = (αx)y = x(αy)

for all α ∈ F and all x, y ∈ A (it is assumed also that the addition in the ring and the addition in the vector space are the same operation). This axiom does hold in the present case. Another algebra, isomorphic in fact to this one, is the algebra of n × n matrices over F (where n = dim V). Another one is F [X] itself.

Given T ∈ Hom_{F} (V, V), consider the mapping

ϕ_{T} : F [X] → Hom_{F}(V, V), ϕ_{T} (f) = f(T).

This is an F -algebra homomorphism, i.e., preserves both the vector space and ring structures. The kernel of this ring homomorphism, like that of any ring homomorphism, is an ideal, so as F [X] is a PID, ker ϕ_{T} = F [X]μ_{T} for a unique monic polynomial μ_{T}, called by definition the minimal polynomial of T. An equivalent definition would be: μ_{T} ∈ F [X] is the minimal polynomial of T if and only if μ_{T} (T) = 0 and μ_{T} is the unique monic polynomial of least degree satisfying this equation. A further property is that if f ∈ F [X] is any polynomial such that f(T) = 0, then μ_{T} | f (in F [X]). Similarly we can define the minimal polynomial of a square matrix A, or of any element x of an F -algebra (in the last case, there is the possibility that μ_{x} = 0, i.e., f(x) ≠ 0 for every nonzero polynomial f ∈ F [X]).

The minimal polynomial of a linear transformation is of course equal to the minimal polynomial of any matrix representing it. This implies that similar matrices have the same minimal polynomial (a fact which is not too hard to prove directly anyway).

The relationship between minimal polynomial and the invariants of the last sections is:

Proposition. Let A be a square matrix over a field F. The minimal polynomial of A is the invariant factor of A of largest degree.

The identical statement is also valid for linear transformations: V → V, V a finitedimensional vector space over F.

Proof. We give the proof for a linear transformation T : V → V. Let f_{1}, …, f_{r} be the invariant factors of T. Thus by definition each f_{i} is monic and

V_{T} ≅ R/f_{1}R ⊕ ⋯ ⊕ R/f_{r}R and f_{1} | f_{2} | ⋯ | f_{r}.

Also by definition, μ_{T} is the unique monic polynomial of least degree such that μ_{T} V_{T} = 0, this equation being equivalent to μ_{T} (T) = 0. Since each f_{i} divides f_{r}, we have f_{r}R ⊆ f_{i}R for each i and so f_{r}V_{T} = 0. Therefore μ_{T} | f_{r}. Conversely μ_{T} (R/f_{r}R) = 0 and so μ_{T} · 1 ≡ 0 mod f_{r}, that is, f_{r} | μ_{T}. Therefore f_{r} = μ_{T}.

Thus the minimal polynomial of T is analogous to the exponent of a finite abelian group.

The following observations are easy to check:

Proposition.

a) If A is in block diagonal form,

A = [ \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} ] ,

then μ_{A} = lcm(μ_{A_{1}}, μ_{A_{2}}).

b) μ_{C(f)} = f if f is monic. (The previous proposition has actually verified this.)

c) μ_{J(α,n)} = (X − α)^{n}. (Again, this follows from the previous proposition, or more messily from a matrix calculation.)

Consequently, for a matrix A, if the Jordan canonical form of A has Jordan blocks J(α_{1}, n_{1}), …, J(α_{r}, n_{r}), then

μ_{A} = lcm((X − α_{1})_{1}^{n}, …, (X − α_{r})_{r}^{n} = ∏_{α}(X − α)^{n_{α}}

where the product is over the distinct diagonal entries α of the Jordan form of A, and for each α, n_{α} is the size of the largest Jordan block appearing for α.

Thus for example, a complex matrix with minimal polynomial (X^{2} − 1)(X^{3} − 1) = (X − 1)^{2}(X + 1)(X − ω)(X − ω^{2}) has Jordan blocks J(−1, 1), J(ω, 1), J(ω^{2}, 1), at least one of each; at least one Jordan block J(1, 2), and possibly some J(1, 1)’s.

This also gives the following characterization of diagonalizability:

Proposition. A square matrix A over an algebraically closed field is diagonalizable, i.e., similar to a diagonal matrix, if and only if its minimal polynomial is square-free, i.e., has no multiple roots (i.e. no repeated linear factors).

The same can be said for a linear transformation T : V → V; we define diagonalizability of such a transformation to mean that there exists a basis v_{1}, …, v_{n} of V and scalars α_{i} ∈ F such that T (v_{i}) = α_{i}v_{i} for each i = 1, …, n.

551, Fall '00      B2e.    Characteristic polynomial; eigenvalues      p. 9

2e. Characteristic polynomial; eigenvalues

We assume familiarity with the rudimentary theory of determinants; every square matrix A over a commutative ring R has a determinant det A ∈ R, which can be obtained by row or column expansion. Moreover,

a) det(AB) = det(A) det(B),

b) det(I) = 1, and

c) A adj A = adj AA = det AI, where adj A is the transpose of the matrix of cofactors: (adj A)_{ij} = (−1)^{i+j} det M_{ji}, where M_{ji} is obtained from A by crossing out the i-th row and j-th column.

d) The matrix A is invertible if and only if det A ≠ 0. (In one direction compute det(AA^{−1}), using a) and b); in the other direction use c).)

e) det [ \begin{matrix}A & 0 \\ B & C \end{matrix} ] = det A det C.

Using determinants we get another useful invariant of a square matrix or linear transformation (maybe the most useful), the characteristic polynomial.

Definition. Let A be an n × n matrix over a commutative ring F. The characteristic polynomial of A is defined to be

χ_{A}(X) = det(XI − A).

The determinant on the right is evaluated in the polynomial ring F [X]. Some authors define it as det(A − XI). This is an inconsequential difference; what matters is just the association class of χ_{A}, and that one sticks consistently to one definition or the other.

Salient properties of characteristic polynomials:

a) If A and A′ are similar then χ_{A} = χ_{A′}.

b) If A is n × n, then χ_{A}(X) = X^{n} − (Tr A)X^{n−1} + ⋯ + (−1)^{n} det A. Here Tr A, the trace of A, is defined as ∑_{i=1}^{n} A_{ii}, the sum of the entries of A on the main diagonal.

c) A is invertible if and only if χ_{A}(0) ≠ 0. (This is immediate since χ_{A}(0) = ± det A.)

d) μ_{A} | χ_{A}. That is, χ_{A}(A) = 0. This is the Cayley-Hamilton Theorem.

Proof. (of Cayley-Hamilton) (proof number 1, 066) Let V = F^{n} be the space of column vectors, with entries in F; let v_{1}, …, v_{n} be the standard basis of V (i.e., the columns of the identity matrix). Make V an F [X]-module in the usual way: for an arbitrary f(X) = a_{n}X^{n} + ⋯ + a_{1}X + a_{0} ∈ F [X], define fv = f(A)v = a_{n}A^{n}v + ⋯ + a_{1}Av + a_{0}v.

Given any matrix C = [c_{ij}(X)] over F [X] (we shall presently take C = XI − A), there is an F -homomorphism ψ_{C} : V → V defined by ψ_{C}(v_{j}) = ∑_{i} c_{ij}(A)v_{i}, j = 1, …, n. (To define homomorphisms out of free modules, precisely what is required is to specify the images of the basis elements.) The following facts follow directly from the definition:

1) If C = fI for some polynomial f ∈ F [X] then ψ_{C} is left multiplication by f(A). (ψ_{C}(v_{j}) = f(A)v_{j}.)

2) ψ_{CD} = ψ_{C} ∘ ψ_{D} for any n × n matrices C and D over F [X].

3) If C = XI − A, then ψ_{C} = 0. (Namely, ψ_{C}(v_{j}) = ∑_{i} c_{ij}(A)v_{i} = Av_{j} − ( ∑_{i} a_{ij}v_{i}) = 0.)

Now let D = adj(XI − A), another n × n matrix over F [X]. Then CD = det(X − IA)I = χ_{A}I, all being matrices over F [X]. By a), ψ_{CD} is left multiplication by χ_{A}(A). But by b) and c), ψ_{CD} = ψ_{C}ψ_{D} = 0. Therefore χ_{A}(A) = 0. QED

Exercise. If A is an n × n matrix over F and A^{m} = 0 for some m, then A^{n} = 0.

Compare the following properties of χ_{A} with those of μ_{A} in the previous section.

Proposition. If

a) If A is in block diagonal form,

A = [ \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} ] ,

then χ_{A} = χ_{A_{1}}χ_{A_{2}}.
b) χ_{C(f)} = f if f is monic.
c) χ_{J(α,n)} = (X − α)^{n}.

Part a) follows from property (e) of determinants, above. For b) and c), you can calculate directly, or be sneaky: we know that these are the minimal polynomials, and since they have the right degree and divide the characteristic polynomials, and every polynomial in sight is monic, they must be the characteristic polynomials. Consequently:

Proposition. χ_{A} is

a) the product of all the invariant factors of A;

b) the product ∏_{α}(X − α)^{m_{a}}, the product being taken over all distinct diagonal entries of the Jordan form of A, with m_{α} being the number of times α appears there (in the case that F is algebraically closed).

Corollary. Let T be a linear transformation V → V. Then μ_{T} = χ_{T} if and only if V_{T} is “cyclic”, i.e., generated as F [X]-module by a single element.

Exercise. Let T : V → V and let W be a T -invariant subspace of V. If there is no w ∈ W such that the images v, T v, T^{2}v, … generate W, then there is no v ∈ V such that the images v, T v, T^{2}v, … generate V. (Hint. Show that the negations of these assertions are that V_{T} and W_{T |W} are cyclic R-modules in the sense just defined.)

Corollary. Any irreducible factor of χ_{A} also divides μ_{A}.

Proof. χ_{A} = f_{1} ⋯ f_{r} with f_{1} | f_{2} | ⋯ | f_{r} = μ_{A}.

Corollary. If A (or T) has at most two invariant factors, then μ_{A} and χ_{A} determine the similarity class of A (or the isomorphism class of V_{T}).

Namely, the invariant factors are μ_{A} and χ_{A}/μ_{A} (the latter existing only if μ_{A} ≠ χ_{A}).

QED

Exercise. Show that χ_{A} and μ_{A} determine the similarity class of A if A is 3 × 3, but not in general if A is 4 × 4 or larger.

Exercise. If A is a complex 4 × 4 matrix and μ_{A} = (X^{2} + 1)(X − i), then what are the possibilities for χ_{A}? Give an example to show that each possibility occurs.

The equation χ_{A}(X) = 0 is the characteristic equation and its roots are the eigenvalues of A. When each eigenvalue α is counted with its multiplicity as a root of the characteristic equation (i.e., the number of linear factors (X − α) of χ_{A}) there are n eigenvalues in all (in an algebraically closed field containing F). The eigenvalues of T are defined similarly.

Proposition. α is an eigenvalue of A (or T) if and only if there is a nonzero column vector v such that Av = αv (or a nonzero v ∈ V such that T (v) = αv).

Proof. χ_{A}(α) = 0 if and only if det(αI − A) = 0 if and only if αI − A is not invertible if and only if there exists v such that (αI − A)v = 0. QED

Such vectors are called eigenvectors; for a fixed α, the eigenvectors together with 0 form a subspace (“eigenspace”).

Exercise. If F is algebraically closed, then the multiplicity of an eigenvalue α is at least the dimension of the eigenspace corresponding to α. The multiplicity of α is exactly equal to the dimension of the generalized eigenspace

V_{α} = ∪_{n=1}^{∞} ker(T − α)^{n}.

A square matrix is nilpotent if and only if its only eigenvalue is 0.

As an example, if A is a 5 × 5 complex matrix which is nilpotent (A^{n} = 0 for some n), then χ_{A} = X^{5}. The largest Jordan block has size m, where μ_{A} = X^{m}. For each of the values m = 5, 4 and 1 there is only one similarity class possible for A: J(0, 5); block(J(0, 4), J(0, 1)), and 0, respectively. For each of the values m = 3, 2 there are two similarity classes possible: block(J(0, 3), J(0, 2)) and block(J(0, 3), J(0, 1), J(0, 1)) for m = 3, and for m = 2 there are either one or two blocks of size 2, and the rest are of size 1.

Exercise. The number of isomorphism classes of abelian groups of order 1024 is the same as the number of similarity classes of 10 × 10 complex matrices. Please do not calculate this number!

Exercise. If A is a nilpotent matrix, then Tr A^{n} = 0 for all n ≥ 0. Show that the converse is false. (Look around: it’s true over some fields!)

Exercise. If T : V → V and W is a T -invariant subspace of V, then μ_{T |W} | μ_{T} and χ_{T |W} | χ_{T}.

Exercise. If A and B are diagonalizable matrices and AB = BA, then they are “simultaneously” diagonalizable, that is, there exists D such that both DAD^{−1} and DBD^{−1} are diagonal.