The characteristic polynomial and determinant are not ad hoc constructions

Skip Garibaldi^{∗}

30 April 2004

1 INTRODUCTION.

Most people are first introduced to the characteristic polynomial and determinant of a matrix in a linear algebra course as undergraduates. The determinant is usually defined as an alternating sum of products of entries of the matrix (as in Jacobi [16, sec. 4]) or as the unique map M_{n}(F) → F that is multilinear and alternating in the columns and has the value 1 at the identity matrix (as in Weierstrass [28] and the books by Hungerford [15], Lang [22], and Dummit and Foote [11]). As a student, I thought that these definitions were at best magical and at worst ad hoc. Where did the determinant come from? This paper gives definitions that I hope the reader will find more natural.

Admittedly, the determinant of a linear transformation on ℝ^{n} is a natural enough object: its absolute value gives the factor by which the transformation enlarges volumes, and its sign says whether or not the map preserves orientation. These properties imply Weierstrass’s axioms (see, for example, [13] or [23, sec. 5]).

Another good definition of the determinant—not so common at the undergraduate level—is in terms of the nth exterior power ∧^{n}F^{n}, as in [5, chap. 3, sec. 8]. This also leads to the Weierstrass axioms.

But even these two “good” definitions have a taint of being special to matrices. (The first is even limited to matrices with real entries.) After all, analogues of the determinant are known for the quaternions, the octonions, finitedimensional field extensions, … It is not clear how to adapt the two good definitions to handle these algebras. As mathematicians, we should demand a definition that works simultaneously in all cases. We give such a definition of the characteristic polynomial in section 2; the constant term of this characteristic polynomial gives an analogue of the determinant. (One normally begins with a definition for the determinant and then defines the characteristic polynomial of a matrix a as det(xI − a). We work in the opposite direction here.) For nby-n matrices, we derive Jacobi’s alternating sum formula for the determinant. We also recover the known ad hoc formulas for the determinant for quaternions and finite-dimensional field extensions. Moreover, the product formula det(aa′) = det(a)det(a′) always holds.

∗This article was published in the American Mathematical Monthly 111, no. 9 (2004), 761–778. This version is very similar to the published version.

The philosophy is the following. Consider the lines in ℝ^{2} given by the equations

ax + by = c, a′x + b′y = c′.

If the coefficients a, b, c, a′, b′, and c′ are specific real numbers, the lines might be parallel or the same (degenerate case), but “typically” they intersect at exactly one point. If we treat the coefficients as independent indeterminates, we say that the lines are generic. Such lines intersect at the point

(x, y) = (\frac{b′c − c′b}{ab′ − a′b}, \frac{ac′ − a′c}{ab′ − a′b}) .

A typical n-by-n matrix has n distinct eigenvalues.^{1} This is true in particular for a generic matrix γ, meaning one whose entries are indeterminates. For such a matrix, the traditional characteristic polynomial is just the minimal polynomial. To define the characteristic polynomial of a specific matrix a, we first find the minimal polynomial minpoly_{γ}(x) of our generic matrix. Plugging in specific values for the indeterminates in γ, we get a polynomial whose only indeterminate is x, and this is the characteristic polynomial of a. This method of defining the characteristic polynomial works for all finite-dimensional F -algebras, and the determinant is (up to a sign) the constant term of the characteristic polynomial.

The core of the idea—looking at the minimal polynomial of a generic element— goes back to the late 1800s (see, for example, [27, p. 241] and [26, p. 301]). All treatments that I have found, however, do not develop the properties of the general characteristic polynomial (as in [9, chap. 7]) or make use of known properties of the characteristic polynomial and determinant for matrices in studying the general characteristic polynomial (as in [1, sec. 10.3], [19], [18], and [17, sec. 5.18]). We use only elementary properties of matrices from the very nice paper [3] and the book [4].

Readers with an algebraic background may argue that one can obtain the characteristic polynomial of an n-by-n matrix over an arbitrary field F by applying the structure theory for finitely-generated torsion modules over a principal ideal domain (as is done in [14, sec. 6.7]). But if one is using that much algebra, the contents of this paper are not so far away and the results here are much stronger.

1If you randomly choose an n-by-n real matrix, the probability that you pick one with distinct complex eigenvalues is 100%. This is because the other real matrices are a set with Lebesgue measure zero. Topologically, amongst the n-by-n matrices, those with distinct eigenvalues form a dense open subset. Over ℝ or ℂ this is true in the usual topology; it is also true over any infinite field in the Zariski topology (see, for example, [25]).

2 THE CHARACTERISTIC POLYNOMIAL.

In this section, we define the characteristic polynomial of an element a in a finite-dimensional F -algebra A and give some of its basic properties. We begin with a definition.

Definition 2.1. Let F be a field. An F -algebra is a ring A with a multiplicative identity 1 (≠ 0) such that A is an F -vector space and

α(ab) = (αa)b = a(αb)  (α ∈ F; a, b ∈ A).  (2.2)

(Alternately, A is a ring with identity such that there is a one-to-one homomorphism F → Z(A) that maps the identity in F to the identity in A.) All algebras that we consider will be finite-dimensional as vector spaces over F. Principal examples of F -algebras are the n-by-n matrices M_{n}(F) and Hamilton’s quaternions, which form an ℝ-algebra (see Example 4.1).

Note that the definition gives a copy of F inside the center of A, but there is no requirement that F be the entire center of A. For example, A may be taken to be a finite-dimensional field extension of F.

For clarity of exposition, we require that A be associative (for example, A cannot be the octonions). However, the definition of the characteristic polynomial that we present also works for the much broader class of strictly powerassociative F -algebras, which includes the octonions and Jordan algebras (see section 7 for precise statements).

The main tool we need that may not be typically discussed in first-year graduate algebra is the tensor product ⊗. It is a canonical way to take a vector space V over the field F and produce a vector space over a larger field K. The new vector space is denoted by K ⊗ V. Heuristically, one thinks of the elements of K ⊗ V as finite sums ∑ k_{i}v_{i} where the k_{i} are in K and the v_{i} are in V. The addition and scalar multiplication are the obvious ones. Formally, one writes such an element as ∑ k_{i} ⊗ v_{i}, although typically we will omit the symbol ⊗. The map v ↦ 1 ⊗ v identifies V with a subset of K ⊗ V. In particular, linearly independent elements of V are sent to linearly independent elements of K ⊗ V. Moreover, if v_{1}, v_{2}, …, v_{m} span the F -vector space V, then their images span the K-vector space K ⊗ V. Consequently, the dimension of V (over F) is the same as the dimension of K ⊗ V (over K).

If the vector space V has a multiplication—as is the case for the algebra A—then so does K ⊗ V. For simple elements k ⊗ v and k′ ⊗ v′ of K ⊗ V, put

(k ⊗ v)(k′ ⊗ v′) = kk′ ⊗ vv′

For general elements, expand (∑ k_{i} ⊗ v_{i})(∑ k′_{j} ⊗ v′_{j}) using the distributive law and apply the rule for simple elements. In this way, K ⊗ A is a K-algebra. For example, K ⊗ M_{n}(F) is isomorphic to M_{n}(K).

(The previous two paragraphs are sufficient for understanding almost all of this paper, but the proofs of Lemmas 2.3 and 6.3 and Propositions 2.7 and 2.11 require a more general version. Let R be an integral domain containing F. We write R ⊗ A for the ring of finite sums ∑ r_{i} ⊗ a_{i} with r_{i} in R and a_{i} in A, endowed with the product described in the preceding paragraph. It is naturally a subring of K ⊗ A for K the field of fractions of R. It is a free R-module whose rank is the dimension of A over F. The curious reader can find a detailed and more general introduction to the tensor product in [11, sec. 10.4].)

Now let a_{1}, a_{2}, …, a_{m} be an F -basis for A, let R = F [t_{1}, t_{2}, …, t_{m}] for (commuting) indeterminates t_{1}, …, t_{m}, and let K be the quotient field of R. We call the element γ = ∑_{i} t_{i}a_{i} of K ⊗ A a generic element. The powers 1, γ, γ^{2}, … of γ live in the finite-dimensional K-vector space K ⊗ A, so they are linearly dependent. That is, there is a nonzero polynomial f(x) in K[x] such that f(γ) = 0. Let minpoly_{γ/K} denote the nonzero monic polynomial in K[x] of smallest degree such that minpoly_{γ/K}(γ) = 0. It is called the minimal polynomial of γ over K.

Note that this polynomial is unique, for if f(x) and g(x) are monic polynomials of minimal degree such that f(γ) = g(γ) = 0, then h(x) = f(x) − g(x) is a polynomial of smaller degree such that h(γ) = 0. This contradicts the minimality of f and g unless h(x) = 0.

Lemma 2.3. The minimal polynomial minpoly_{γ/K} is in R[x], not just K[x].

Proof. Consider the R-submodules A_{j} of R ⊗ A generated by {1, γ, γ^{2}, …, γ^{j}}. They form an ascending chain A_{1} ⊆ A_{2} ⊆ · · ·. Since R is Noetherian (Hilbert’s Basis Theorem) and R ⊗ A is a finitely-generated R-module, this chain must stabilize. That is, γ^{j+1} is in A_{j} for some j, so γ satisfies a monic polynomial f in R[x].^{2} Since minpoly_{γ/K} divides f in K[x] and both are monic, minpoly_{γ/K} lies in R[x] by Gauss’s lemma.

Definition 2.4. Express the element a of A in terms of the basis a_{1}, a_{2}, …, a_{m} as a = ∑_{i} α_{i}a_{i} with α_{i} in F. The substitution t_{i} ↦ α_{i} defines a map R[x] → F [x]. We call the image of minpoly_{γ/K} in F [x] the characteristic polynomial of a and denote it by chpoly_{a,A/F} or simply chpoly_{a}.

Remark 2.5. It is immediate from the definition that

deg(chpoly_{a,A/F}) ≤ dim_{F} A

and that the degree of chpoly_{a,A/F} is the same for all a in A.

Example 2.6 (Upper-triangular matrices). Let A be the algebra of nby-n upper-triangular matrices over F. Write E_{ij} for the matrix whose only nonzero entry is a 1 in the (i, j)-position. Fix a basis a_{1}, a_{2}, …, a_{m} for A over F consisting of E_{ij} such that a_{i} = E_{ii} when 1 ≤ i ≤ n. Let γ be the corresponding generic element defined earlier.

2A more direct argument would be: R[γ] is an R-submodule of R⊗A and R⊗A is a finitelygenerated R-module. Hence γ is integral over R [15, Theorem 8.5.3]. Unfortunately, the typical proof of this implication invokes determinants, so we use instead that R is Noetherian.

Let I_{n} denote the n-by-n identity matrix. For i = 1, 2, …, n the matrix γ − t_{i}I_{n} has n − 1 pivot columns—equivalently, n − 1 leading 1s—in its rowreduced form, hence it has a nonzero kernel. That is, γ has an eigenvector in K^{n} with eigenvalue t_{i}. Since the t_{i} are distinct elements of K, these eigenvectors form a basis for K^{n}, so γ is similar in M_{n}(K) to the diagonal matrix with diagonal entries t_{1}, t_{2}, …, t_{n}. The minimal polynomial of γ is ∏_{i=1}^{n}(x − t_{i}), since similar matrices have the same minimal polynomials.^{3} By substitution, an upper-triangular matrix b has characteristic polynomial ∏_{i=1}^{n}(x − b_{ii}).

We now show that the characteristic polynomial is well defined in general.

Proposition 2.7. The characteristic polynomial chpoly_{a,A/F} depends only on a, A, and F (and not on the choice of basis a_{1}, a_{2}, …, a_{m} for A).

Proof. Suppose that we take another F -basis b_{1}, b_{2}, …, b_{m} of A with a corresponding generic element ε = ∑_{i} t_{i}b_{i}. We may write b_{i} = ∑_{j} g_{ij}a_{j} for g an invertible matrix in M_{m}(F). Let f : R → R be the F -algebra automorphism defined by

f(t_{j}) = ∑_{i} t_{i}g_{ij}.

Write a in A in terms of each basis as

a = ∑_{j} α_{j}a_{j} = ∑_{i} β_{i}b_{i}.  (2.8)

We have a diagram

R[x] ⟶^{t_{j}↦α_{j}} F [x]
f↓  ∥
R[x] ⟶^{t_{i}↦β_{i}} F [x]

with horizontal arrows the substitution maps. Equation (2.8) ensures that α_{j} = ∑_{i} β_{i}g_{ij} for all j, hence the diagram commutes.

If we begin with minpoly_{γ/K} in the upper left, substitution gives the member of F [x] that is chpoly_{a} computed with respect to the basis a_{1}, a_{2}, …, a_{m}. On the other hand, f extends in an obvious way to an automorphism of R ⊗ A such that

f(γ) = ∑_{j} f(t_{j})a_{j} = ∑_{j} ( ∑_{i} t_{i}g_{ij})a_{j} = ∑_{i} t_{i}( ∑_{j} g_{ij}a_{j}) = ε.

Hence f(minpoly_{γ/K}) = minpoly_{ε/K}. The image of this in F [x] is chpoly_{a} computed with respect to the basis b_{1}, b_{2}, …, b_{m}. The commutativity of the diagram establishes the proposition.

3This argument may appear to be excessively long. It is included here to illustrate that we are not making use of determinants.

Undergraduates are asked to find the characteristic polynomial of a matrix whose entries are specific rational numbers. Of course their answer is the same whether they think of the matrix as living in M_{n}(ℚ), M_{n}(ℝ), or M_{n}(ℂ). The characteristic polynomial from Definition 2.4 has the same property.

Lemma 2.9. Let E be a field containing F, and fix a in A. The minimal polynomials and characteristic polynomials of a are the same over F and over E.

Proof. Let d be the degree of the polynomial minpoly_{a/F}. The elements 1_{A}, a, …, a^{d−1} are linearly independent over F, hence the elements 1 ⊗ 1_{A}, 1 ⊗ a, …, 1 ⊗ a^{d−1} of E ⊗A are linearly independent over E. Note that 1⊗a^{i} equals (1⊗a)^{i} for all i, so the minimal polynomial minpoly_{(1⊗a)/E} of a over E has degree at least d. Since this polynomial divides minpoly_{a/F}, the two polynomials are the same.

The F -basis a_{1}, a_{2}, …, a_{m} of A gives an E-basis 1 ⊗ a_{1}, 1 ⊗ a_{2}, …, 1 ⊗ a_{m} of E ⊗ A, and the generic element constructed from this E-basis is the image of γ in E(t_{1}, t_{2}, …, t_{m}) ⊗ A. Since the minimal polynomials of γ over K and over E(t_{1}, …, t_{m}) are the same by the preceding paragraph, we get

chpoly_{a,A/F} = chpoly_{(1⊗a),(E⊗A)/E}

by substitution.

In general, we write

chpoly_{a}(x) = x^{n} − c_{1}(a)x^{n−1} + ⋯

⋯ + (−1)^{n−1}c_{n−1}(a)x + (−1)^{n}c_{n}(a). (2.10)

The elements c_{1}(a) and c_{n}(a) play the roles of the trace and determinant of a.

Proposition 2.11. Let A be a finite-dimensional F -algebra. Then the following statements are true:

(1) (Cayley-Hamilton) chpoly_{a}(a) = 0 for each a in A.

(2) If φ: A → A is a ring automorphism or anti-automorphism that restricts to an automorphism of F, then φ(c_{i}(a)) = c_{i}(φ(a)) for all a in A.

(3) The c_{i} satisfy c_{i}(αa) = α^{i}c_{i}(a) for all α in F and a in A.

(4) The mapping c_{1} : A → F is F -linear.

(5) If B is a subalgebra of A and b is in B, then chpoly_{b,B/F} divides chpoly_{b,A/F} in F [x].

We will observe in Theorem 3.4 that our notion of characteristic polynomial on M_{n}(F) is the same as the usual one. Then Proposition 2.11 contains many results that one typically proves in a linear algebra course. For example, (2) tells us that that similar matrices have the same characteristic polynomial and that det(a^{t}) = det(a) for a in M_{n}(F), where a^{t} denotes the transpose of a.

Proof. To establish (1), write a = ∑_{i} α_{i}a_{i}. Then chpoly_{a}(a) is obtained by making the substitution t_{i} ↦ α_{i} in minpoly_{γ/K}(γ). (This substitution defines a homomorphism R ⊗ A → A.) Since minpoly_{γ/K}(γ) = 0 in R ⊗ A, we have chpoly_{a}(a) = 0 in A.

We next prove (2). The map φ extends naturally to an automorphism g ↦ φg of K[x] by applying φ to the coefficients of g. Similarly, φ extends to an automorphism or anti-automorphism of R ⊗ A such that ^{φ}g(φ(u)) = φ(g(u)) for every g in R[x] and u in R ⊗ A. We have: g(γ) = 0 if and only if ^{φ}g(φ(γ)) = 0. Hence

φminpoly_{γ/K} = minpoly_{φ(γ)/K} .

The diagram

R[x] ⟶^{t_{i}↦α_{i}} F [x]
φ↓  ↓^{φ}
R[x] ⟶^{t_{i}↦φ(α_{i})} F [x]

commutes. Beginning with minpoly_{γ/K} in the upper left and going clockwise, we obtain chpoly_{a} in the upper right F [x], and then ^{φ}chpoly_{a} in the lower right F [x]. Going counterclockwise, we obtain ^{φ}minpoly_{γ/K} = minpoly_{φ(γ)/K} in the lower left. The image in the lower right F [x] is the characteristic polynomial of φ(a) (computed with respect to the basis φ(a_{1}), φ(a_{2}), …, φ(a_{m}) of A, but that is irrelevant by Proposition 2.7). The commutativity of the diagram gives the desired equality chpoly_{φ(a)} = ^{φ}chpoly_{a}.

Turning to (3) and (4), we suppose first that α is not 0. If we write the minimal polynomial of γ/K as ∑_{i=0}^{n} c_{i}x^{i} for c_{i} in R, then the minimal polynomial of αγ/K is ∑_{i=0}^{n} c_{i}α^{n−i}x^{i}. Thus

α^{n} minpoly_{γ/K}(x) = minpoly_{αγ/K}(αx) (in R[x])

and

α^{n} chpoly_{a}(x) = chpoly_{αa}(αx) (in F [x]).

This gives

α^{n}c_{i}(a) = α^{n−i}c_{i}(αa)  (0 ≤ i ≤ n; α ≠ 0).  (2.12)

In particular, (2.12) holds when α is an indeterminate. Since c_{i}: A → F is given by a polynomial in the coordinates of a with respect to some basis a_{1}, a_{2}, …, a_{m}, this polynomial is homogeneous of degree i. This gives (3) and (4).

For (5), fix a basis b_{1}, b_{2}, …, b_{r} of B and extend it to a basis a_{1}, a_{2}, …, a_{m} of A with a_{i} = b_{i} when 1 ≤ i ≤ r. By analogy, set S = F [t_{1}, t_{2}, …, t_{r}], let L denote the quotient field of S, and let ε be the generic element ∑_{i=1}^{r} t_{i}b_{i} in S ⊗ A.

Write b = ∑_{i} β_{i}b_{i} with β_{i} in F. Define a map ϕ: R → S by sending t_{j} to 0 when r < j ≤ m. The image of minpoly_{γ/K} under the composition

R[x] ⟶^{ϕ} S[x] ⟶^{t_{i}↦β_{i}} F [x]

is chpoly_{b,A/F}. Similarly, the image of minpoly_{ε/L} is chpoly_{b,B/F}.

The homomorphism ϕ extends naturally to a map R ⊗ A → S ⊗ A such that ϕ(γ) = ε. We have

0 = ϕ(0) = ϕ(minpoly_{γ}(γ)),

which equals the polynomial ϕ(minpoly_{γ}) in S[x] evaluated at ϕ(γ) = ε. Consequently, minpoly_{ε/L} divides ϕ(minpoly_{γ/K}) in L[x]. Since S is a unique factorization domain, minpoly_{ε/L} divides ϕ(minpoly_{γ/K}) in S[x]. Consequently, the image chpoly_{b,B/F} of minpoly_{ε/L} in F [x] divides the image chpoly_{b,A/F} of ϕ(minpoly_{γ/K}).

3 MATRICES.

In this section, we observe that the characteristic polynomial as defined in the previous section agrees with the usual linear algebra notion of characteristic polynomial in the case where A = M_{n}(F).

Everyone knows the next lemma, but maybe not the clean proof:

Lemma 3.1. If T is a linear transformation on an F -vector space of positive dimension n, then T satisfies a nonzero polynomial of degree at most n.

Proof. We sketch the nice proof from [7] that proceeds by induction on n. The case n = 1 is clear. We assume that n > 1. Let v be a nonzero vector in the vector space V. The n + 1 vectors

v, T (v), T^{2}(v), …, T^{n}(v)

must be linearly dependent, so there is a nonzero polynomial g(x) in F [x] of degree ≤ n such that g(T)v = 0.

Set U = ker g(T). The linear transformations T and g(T) commute, so T (U) ⊆ U and T induces a linear transformation T_{V/U} on V/U. By induction, T |_{U} satisfies a polynomial m_{U}(x) with deg(m_{U}) ≤ dim U, and T_{V/U} satisfies a polynomial m_{V/U}(x) with deg(m_{V/U}) ≤ dim V/U = dim V − dim U. Then m_{V/U}(T)V ⊆ U and T satisfies the polynomial m_{U}(x) m_{V/U}(x). Moreover,

deg(m_{U} · m_{V/U}) = deg(m_{U}) + deg(m_{V/U}) ≤ dim U + dim(V/U) = dim V.

Corollary 3.2. The characteristic polynomial (in the sense of this paper) of a matrix in M_{n}(F) has degree at most n.

Proof. Lemma 3.1 applies in particular to an element γ of M_{n}(K) considered as a linear transformation of K^{n}. Substitution gives the corollary.

Fix an algebraic closure \overline{F} of F. For a in M_{n}(F), we call λ in \overline{F} an eigenvalue of a if the kernel of (λI_{n} − a) is nonzero. Let U_{λ} denote the corresponding generalized eigenspace; i.e., U_{λ} is the set of vectors v in \overline{F}^{n} that belong to the kernel of (λI_{n} − a)^{r} for some natural number r. The multiplicity m(λ) of an eigenvalue λ is dim\overline{F} U_{λ}.

Example 3.3 (Upper-triangular matrices). The multiplicity of an eigenvalue λ of an upper-triangular matrix b is the number of times λ appears as a diagonal entry of b. This can be proved directly from the definitions (see, for example, [4, Theorem 8.10]).

Theorem 3.4. For a in M_{n}(F) the characteristic polynomial of a (as in Definition 2.4) factors in \overline{F} [x] as

(x − λ_{1})^{m(λ_{1})}(x − λ_{2})^{m(λ_{2})} ⋯ (x − λ_{k})^{m(λ_{k})},  (3.5)

where λ_{1}, λ_{2}, …, λ_{k} are the distinct eigenvalues of a.

Proof. Let b be an upper-triangular matrix, and let B denote the subalgebra of M_{n}(F) consisting of all upper-triangular matrices. The characteristic polynomial chpoly_{b,B/F} of b as an element of B was computed in Example 2.6: it is given by (3.5) as observed in Example 3.3. According to Proposition 2.11(5), chpoly_{b,B/F} divides the characteristic polynomial chpoly_{b,M_{n}(F)/F} of b as an element of M_{n}(F). Since both polynomials are monic and have degree n (Example 2.6, Corollary 3.2), the theorem holds for upper-triangular matrices.

Since the characteristic polynomial of the given matrix a is unchanged under scalar extension, we may assume that F is algebraically closed, i.e., F = \overline{F}. Here we need to invoke one somewhat sophisticated result from linear algebra: since F is algebraically closed, a is similar to an upper-triangular matrix b [3, Theorem 6.2]. But the theorem holds for b by the preceding paragraph. Since similarity changes neither the characteristic polynomial (Proposition 2.11(2)) nor the eigenvalues, the theorem holds for a.

In [3], Axler develops many of the typical properties of matrices (e.g., the existence of eigenvalues and the decomposition with respect to generalized eigenspaces) over an algebraically closed field without use of the determinant. For example, in section 5 of that paper he defines the characteristic polynomial to be exactly the product displayed in the theorem. Logically, one could replace his section 5 with this paper.

Corollary 3.6. For a in M_{n}(F) the minimal polynomial minpoly_{a} and the characteristic polynomial chpoly_{a} have the same irreducible factors in F [x].

Proof. Irreducible polynomials in F [x] are determined (up to scalar factors) by their roots in an algebraic closure \overline{F}. Thus we may assume that F is algebraically closed.

As in the proof of Theorem 3.4, a is similar to an upper-triangular matrix. Since conjugation changes neither the characteristic nor the minimal polynomial, we may assume that a is upper-triangular.

By Theorem 3.4, every irreducible factor of the characteristic polynomial is of the form (x − λ_{i}) where λ_{i} is a diagonal entry in a, say λ_{i} = a_{ii}. The (i, i)- entry of minpoly_{a}(a) = 0 is 0, but it is also minpoly_{a}(λ_{i}). Therefore, (x − λ_{i}) divides minpoly_{a}.

Since minpoly_{a} divides chpoly_{a} by the Cayley-Hamilton theorem (Proposition 2.11(1)), the corollary is proved.

Proposition 3.7. For a and a′ in M_{n}(F), the following statements are true:

(1) a is invertible if and only if c_{n}(a) ≠ 0.

(2) (Jacobi formula) c_{n}(a) = ∑_{σ∈S_{n}}(sign σ)a_{1σ(1)}a_{2σ(2)} ⋯ a_{nσ(n)}.

(3) c_{n}(aa′) = c_{n}(a)c_{n}(a′).

(4) c_{1}(a) = a_{11} + a_{22} + ⋯ + a_{nn}.

Proof. For (1), we note that the matrix a is invertible if and only if the kernel of a is zero [4, Proposition 3.17], which happens if and only if 0 is not an eigenvalue of A. By Theorem 3.4, this is true if and only if c_{n}(a) ≠ 0.

We now follow [3, sec. 9]. Write d(a) for the right-hand side of (2). A straightforward rearrangement of terms as in [6, p. 179] or [4, Theorem 10.31] shows that d(aa′) = d(a)d(a′). Therefore, (2) implies (3).

We now prove (2). Suppose first that a is upper-triangular. Then both sides of (2) are just the product of the diagonal entries of a, hence (2) holds in this case.

Now consider the general case. Since c_{n}(a) and d(a) are unchanged if we enlarge our base field, we may assume that F is algebraically closed, in which event a is similar to an upper-triangular matrix b, i.e., b = gag^{−1} for some g in M_{n}(F). Then

c_{n}(b) = d(b) = d(gag^{−1}) = d(g)d(ag^{−1}) = d(ag^{−1})d(g) = d(ag^{−1}g) = d(a).

Since c_{n}(a) = c_{n}(b) by Proposition 2.11(2), we have proved (2).

As to (4), by Proposition 2.11(4) there exist β_{ij} in F such that

c_{1}(a) = ∑_{i} ∑_{j} β_{ij}a_{ij}

for every matrix a. In the case where a is upper-triangular, the proof of Theorem 3.4 shows that the characteristic polynomial of a is as in Example 2.6. In particular, (4) holds. This gives

β_{ij} = \begin{cases}0 & if i < j, \\ 1 & if i = j. \end{cases}

A symmetric argument with lower-triangular matrices shows that β_{ij} = 0 when i > j.

Definition 3.8. For a an element of a finite-dimensional F -algebra A, we define the trace of a to be

tr_{A/F} (a) = c_{1}(a)

and the determinant of a to be

det_{A/F} (a) = c_{n}(a).

If there is no danger of ambiguity, we write simply det_{A} or det instead of det_{A/F} and similarly for the trace. The trace tr_{M_{n}(F)} and determinant det_{M_{n}(F)} are the usual trace and determinant from linear algebra by Proposition 3.7—or by Theorem 3.4, depending on your point of view.

4 QUATERNIONS.

Example 4.1. Hamilton’s quaternions—usually denoted by ℍ—are defined to be the ring constructed by taking the complex numbers ℂ and adjoining an element j such that j^{2} = −1, j commutes with real numbers, and ij = −ji. It has ℝ-basis 1, i, j, and k, where k = ij. A lot of interesting information about the quaternions can be found in [12, chap. 7]. (Note that ℍ is an ℝ-algebra but not a ℂ-algebra, since ℂ is not in the center of ℍ.)

Set ϕ to be the ℝ-linear map ℍ → M_{2}(ℂ) defined by

1 ↦ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , i ↦ (\begin{matrix}i & 0 \\ 0 & −i \end{matrix}) , j ↦ (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , k ↦ (\begin{matrix}0 & i \\ i & 0 \end{matrix}) .

This extends to an isomorphism ϕ: ℂ ⊗ ℍ → M_{2}(ℂ) (as ℂ-algebras).

Since the characteristic polynomial is unchanged when we enlarge our base field (Lemma 2.9), the characteristic polynomial of a quaternion q has degree 2, just like a matrix in M_{2}(ℂ). That is,

chpoly_{q}(x) = x^{2} − tr_{ℍ}(q) x + det_{ℍ}(q).

We now determine tr_{ℍ} and det_{ℍ}. Every quaternion can be written as q = r + si + uj + vk = z + wj for some real numbers r, s, u, and v and complex numbers z = r + si and w = u + vi. We have

ϕ(q) = (\begin{matrix}z & w \\ −\overline{w} & \overline{z} \end{matrix}) .

Because the characteristic polynomial is unchanged when we enlarge our base field, we find that

tr_{ℍ}(q) = tr_{M_{2}(ℂ)}(ϕ(q)) = z + \overline{z} = 2r

and

det_{ℍ}(q) = det_{M_{2}(ℂ)}(ϕ(q)) = z\overline{z} + w\overline{w} = r^{2} + s^{2} + u^{2} + v^{2}.

Example 4.2 (Matrices over the quaternions). Write M_{2}(ℍ) for the set of 2-by-2 matrices with entries in ℍ. The obvious addition and multiplication make it into a 16-dimensional ℝ-algebra.

In [8], Cayley defined a determinant Cdet: M_{2}(ℍ) → ℍ by

Cdet (\begin{matrix}q_{11} & q_{12} \\ q_{21} & q_{22} \end{matrix}) = q_{11}q_{22} − q_{21}q_{12}.

He noted that his determinant had some unsavory properties. For example,

Cdet (\begin{matrix}q & q′ \\ q & q′ \end{matrix}) = 0 (q, q′ ∈ ℍ),

whereas

Cdet (\begin{matrix}i & i \\ j & j \end{matrix}) = ij − ji = 2ij ≠ 0.

We can constrast this with the determinant that we have just defined. Just as for ℍ, there is an isomorphism ϕ_{2} : ℂ ⊗ M_{2}(ℍ) → M_{4}(ℂ) such that

ϕ_{2} (\begin{matrix}q_{11} & q_{12} \\ q_{21} & q_{22} \end{matrix}) = (\begin{matrix}ϕ(q_{11}) & ϕ(q_{12}) \\ ϕ(q_{21}) & ϕ(q_{22}) \end{matrix}) .

(Recall that ϕ(q) is a 2-by-2 complex matrix for every q in ℍ.) If m in M_{2}(ℍ) has a repeated row or column, then so does ϕ_{2}(m), hence

det_{M_{2}(ℍ)}(m) = det_{M_{4}(ℂ)}(ϕ_{2}(m)) = 0.

More generally, our trace and determinant have the nice properties of the usual trace and determinant for matrices as given in Proposition 2.11.

For a more comprehensive discussion of various types of determinants for M_{2}(ℍ), see [2]. Aslaksen refers to our det_{M_{2}(ℍ)} as the Study determinant.

Example 4.3 (Central simple algebras). A typical topic for a first-year graduate algebra course is Wedderburn’s description of simple Artinian rings: they are isomorphic to M_{r}(D) for D a skew field. We write F for the center of D (which is necessarily a field), and suppose that D is finite-dimensional over F. Such an algebra M_{r}(D) is called central simple. We have just seen two examples of these, with F = ℝ, D = ℍ, and r = 1, 2.

The trace tr_{M_{r}(D)} and determinant det_{M_{r}(D)} are called the reduced trace and reduced norm, respectively. They are usually constructed by “Galois descent” as in [10, p. 145] or [24], using facts special to central simple algebras. Here we get them as a consequence of the general theory of the characteristic polynomial from section 2.

5 MORE PROPERTIES OF THE CHARAC-TERISTIC POLYNOMIAL.

Here we discuss the example of finite-dimensional field extensions (Example 5.2). We also establish some additional nice properties of the characteristic polynomial (Corollary 5.3).

Write End_{F} (A) for the set of F -linear maps A → A. It is an F -algebra, with function composition as multiplication. It is isomorphic to M_{m}(F).

For a in A, write L_{a} for the element of End_{F} (A) defined by

L_{a}(b) = ab (b ∈ A).

The map a ↦ L_{a} defines an F -algebra homomorphism called the left regular representation of A. This homomorphism is injective: if L_{a} = 0, then L_{a}(a′) = 0 for all a′ in A, whence 0 = L_{a}(1_{A}) = a · 1_{A} = a.

Proposition 5.1. Let a be an element in a finite-dimensional F -algebra A. The minimal polynomial minpoly_{a/F} divides the characteristic polynomial chpoly_{a,A/F} of a, which in turn divides the characteristic polynomial chpoly_{L_{a},End_{F} (A)/F}, all in F [x]. All three polynomials have the same irreducible factors in F [x].

Proof. Since chpoly_{a}(a) = 0 by the Cayley-Hamilton theorem (Proposition 2.11(1)), the minimal polynomial of a divides the characteristic polynomial chpoly_{a}. Because the left regular representation is injective, we see that chpoly_{a} divides chpoly_{L_{a}} by 2.11(5).

We are reduced to showing that chpoly_{L_{a}} and minpoly_{a} have the same irreducible factors. Now the left regular representation is injective, so a and L_{a} have the same minimal polynomials. That is, we need show only that chpoly_{L_{a}} and minpoly_{L_{a}} have the same irreducible factors. Since End_{F} (A) is isomorphic to M_{m}(F) for m = dim_{F} A, an appeal to Corollary 3.6 completes the proof.

The proposition gives us the power to handle another example.

Example 5.2 (Finite-dimensional field extensions). Let A be an extension field of F of finite dimension m. Every element a of A corresponds to an element L_{a} in End_{F} (A) ≅ M_{m}(F), and the characteristic polynomial of a divides the characteristic polynomial of L_{a} by Proposition 5.1. The trace and norm of a are usually defined to be the trace and determinant of L_{a}.

If A is separable over F, then by the Theorem of the Primitive Element A = F [θ] for some θ in A. The minimal polynomial of θ has degree m. Since it divides the characteristic polynomial of L_{θ} by Proposition 5.1 and the latter polynomial has degree m, we find that

chpoly_{θ,A/F} = chpoly_{L_{θ}} .

The characteristic polynomials chpoly_{a} have the same degree for all a in A. Accordingly,

chpoly_{a} = chpoly_{L_{a}} (a ∈ A).

In particular, for finite separable field extensions, our trace and determinant agree with the usual trace and norm.

If A is not separable over F, there can be some disagreement. For example, let F = 𝔽_{2}(u, v) be the field of rational functions in indeterminates u and v, where 𝔽_{2} is the field with two elements. The field A = F (\sqrt{u}, \sqrt{v}) is a purely inseparable extension of dimension 4 over F, with basis 1, \sqrt{u}, \sqrt{v}, and \sqrt{uv}. The generic element

γ = t_{1} · 1 + t_{2}\sqrt{u} + t_{3}\sqrt{v} + t_{4}\sqrt{uv}

has minimal polynomial

γ^{2} − (t_{1}^{2} + t_{2}^{2}u + t_{3}^{2}v + t_{4}^{2}uv).

Here the characteristic polynomial of each element a in A divides but is not equal to the characteristic polynomial of L_{a}.

Proposition 5.1 also allows us to prove that many nice properties of the characteristic polynomial of a matrix hold as well for characteristic polynomials of elements of A.

Recall (Remark 2.5) that the characteristic polynomial chpoly_{a} has the same degree for every a in A. We say that A has degree n if chpoly_{a} has degree n for each a in A.

Corollary 5.3. Let A be a finite-dimensional F -algebra of degree n. For a in A, the following assertions hold:

(1) c_{i}(1_{A}) = \binom{n}{i}. In particular, tr_{A}(1_{A}) = n and det_{A}(1_{A}) = 1.

(2) a is invertible if and only if det_{A}(a) ≠ 0.

(3) a is nilpotent if and only if chpoly_{a} = x^{n}.

Proof. For (1) we observe that the minimal polynomial of 1_{A} is x − 1. Proposition 5.1 thus gives

chpoly_{1_{A}} = (x − 1)^{n}.

To treat the (⇐) direction of (2), we first define the adjoint of a, denoted by adj a, as follows:

adj a = (−1)^{n+1}[a^{n−1} − c_{1}(a)a^{n−2} + ⋯ + (−1)^{n−1}c_{n−1}(a)].

Then

a · adj a = (−1)^{n+1}[chpoly_{a}(a) − (−1)^{n}c_{n}(a)] = det_{A}(a)1_{A}.

This shows that a is invertible when det_{A}(a) is not zero. For the converse, assume that a is invertible. Then L_{a} is invertible with inverse L_{a^{−1}}. By Proposition 3.7(1), the constant term det_{End_{F} (A)}(L_{a}) of chpoly_{L_{a}} is not zero. Because chpoly_{a,A/F} and chpoly_{L_{a},End_{F} (A)/F} have the same irreducible factors in F [x], the constant term det_{A}(a) of chpoly_{a,A/F} is not zero.

Finally, we consider (3). The element a is nilpotent if and only if it satisfies the polynomial x^{r} for some natural number r, which is the case if and only if a has minimal polynomial x^{p} for some natural number p. Since the minimal polynomial and characteristic polynomial have the same irreducible factors (Proposition 5.1), this holds if and only if the characteristic polynomial of a is x^{n}.

Remark 5.4. One might be tempted to accept the traditional definition of characteristic polynomial for matrices, and then define the characteristic polynomial of a in A as chpoly_{L_{a},End_{F} (A)/F}. But this definition is unsatisfactory for the quaternions, because their traditional characteristic polynomial—derived in Example 4.1—has degree two, but chpoly_{L_{a}} would have degree four. And there is another problem: there is no strong mathematical reason to prefer the left regular representation over the right regular representation (defined in the obvious manner as a ↦ R_{a}), and the characteristic polynomials of L_{a} and R_{a} may differ. Adrian Wadsworth points out that the upper-triangular matrices from Example 2.6 provide an example of this difficulty. In particular, the generic element γ has

chpoly_{L_{a}}(γ) = (x − t_{1})^{n}(x − t_{2})^{n−1} ⋯ (x − t_{n−1})^{2}(x − t_{n})

and

chpoly_{R_{a}}(γ) = (x − t_{1})(x − t_{2})^{2} ⋯ (x − t_{n−1})^{n−1}(x − t_{n})^{n}.

6 THE PRODUCT FORMULA FOR DETER-MINANTS.

In this section, we prove that the usual product formula for determinants of matrices holds for an arbitrary finite-dimensional F -algebra A:

Theorem 6.1. For every a and a′ in A, it is true that

det_{A}(aa′) = det_{A}(a) det_{A}(a′).

We postpone the proof until the end of the section. A more cryptic—but also more powerful—proof can be found in [19, chap. 6, sec. 5].

Remark 6.2. Recall that det_{A} is both a function A → F and an element of R = F [t_{1}, …, t_{m}]. To evaluate the function det_{A} at a in A, we write a in terms of the basis a_{1}, a_{2}, …, a_{m} as a = ∑_{i} α_{i}a_{i} and make the substitution t_{i} ↦ α_{i} in the polynomial det_{A}. In this manner, we may view each polynomial in R as a function A → F.

Lemma 6.3. Suppose that F is infinite. Let f in R be such that f(1_{A}) = 1 and f(aa′) = f(a) f(a′) for all a and a′ in A. If a member g of R divides f and has g(1_{A}) = 1, then g(aa′) = g(a) g(a′) for all a and a′ in A.

Proof. Since f(1_{A}) = g(1_{A}) = 1, the polynomials f and g are nonzero elements of R. Suppose that f or g belongs to the group F^{×} of units in R. Then g is a unit, hence g = 1 and the lemma holds.

We may assume that f and g are nonzero nonunits. Since R is a unique factorization domain, we can write f = f_{1}f_{2} ⋯ f_{r}, where each factor f_{k} is irreducible in R. After multiplying the f_{k} by elements of F^{×} if necessary, we may assume that f_{k}(1_{A}) = 1 for every k. The fact that g divides f and g(1_{A}) = 1 means that g is a product g = ∏_{ℓ∈L} f_{ℓ} for some subset L of {1, …, n}. To prove the lemma, it suffices to prove that f_{k}(aa′) = f_{k}(a)f_{k}(a′) for k = 1, …, n and for all a, a′ in A.

Set R′ = F [u_{1}, …, u_{m}, v_{1}, …, v_{m}] for independent indeterminates u_{1}, …, u_{m}, v_{1}, …, v_{m}. Define elements μ and ν of R′ ⊗ A by

μ = ∑_{i} u_{i}a_{i}, ν = ∑_{j} v_{j}a_{j}.

As in Remark 6.2, every element of R defines a map R′ ⊗ A → R′. For example, to find f(μ) we take f and replace t_{i} with u_{i}. Similarly, we get f(ν) via the substitution t_{j} ↦ v_{j}.

We claim that f(μν) = f(μ)f(ν). Let δ = f(μν) − f(μ)f(ν) in R′. Every element of R′ defines a function F^{2m} → F by plugging in for the u_{i} and v_{j}. Making the substitutions u_{i} ↦ α_{i} and v_{j} ↦ β_{j} in δ, where α_{i} and β_{j} belong to F, we obtain

f(ab) − f(a)f(b)

for a = ∑ α_{i}a_{i} and b = ∑ β_{j}a_{j}, which is 0 by hypothesis. That is, δ gives the map F^{2m} → F that is identically 0. Since F is infinite, δ = 0. (This can be proved by induction on the number of variables appearing in the polynomial δ, using that fact that a nonzero polynomial in one variable has only finitely many roots; see [21, Proposition 1.3a].) This establishes the claim.

Thus

∏_{k=1}^{r} f_{k}(μν) = f(μν) = f(μ)f(ν) = ∏_{k=1}^{r} f_{k}(μ)f_{k}(ν)

in R′. Recall that we obtain f_{k}(μ) and f_{k}(ν) by substituting one set of indeterminates for another in f_{k}. Hence, since f_{k} is prime in R, the polynomials f_{k}(μ) and f_{k}(ν) are prime in R′ for all k. Therefore the prime factorization of f_{k}(μν) in the unique factorization domain R′ is a product of f_{p}(μ) and f_{q}(ν) for some ps and qs. Substituting for the v_{j} so that ν is sent to 1_{A} has the effect that

f_{k}(μν) ↦ f_{k}(μ), f_{p}(μ) ↦ f_{p}(μ), f_{q}(ν) ↦ f_{q}(1_{A}) = 1

for all p and q. Hence the only irreducible factor of f_{k}(μν) amongst the f_{p}(μ) is f_{k}(μ). Similarly, substituting for the u_{i} so that μ ↦ 1_{A}, we have f_{k}(μν) ↦ f_{k}(ν). It follows that

f_{k}(μν) = f_{k}(μ)f_{k}(ν) (k = 1, …, n).

By doing the substitution μ ↦ a and ν ↦ a′, we obtain f_{k}(aa′) = f_{k}(a)f_{k}(a′) for all a and a′ in A.

Proof of Theorem 6.1. Since the determinant of an element of A is unchanged when we enlarge the base field (Lemma 2.9), we may assume that F is infinite. Let γ = ∑ t_{i}a_{i} be a generic element of A as in section 2, and consider the element L_{γ} in End_{K}(K ⊗ A).

The minimal polynomial minpoly_{γ} divides chpoly_{L_{γ},End_{K}(K⊗A)/K} in K[x] by Proposition 5.1. By Lemma 2.3, both polynomials actually lie in R[x] for R = F [t_{1}, …, t_{m}], so minpoly_{γ} divides chpoly_{L_{γ}} in R[x] by Gauss’s lemma. Substituting 0 for x defines a surjection R[x] → R that sends minpoly_{γ} to (−1)^{n}det_{A} for n = deg(minpoly_{γ}) and chpoly_{L_{γ}} to the function a ↦ (−1)^{m}det_{End_{F} (A)}(L_{a}). Since substitution is a ring homomorphism, det_{A} divides a ↦ det_{End_{F} (A)}(L_{a}) in R.

We have

det_{End_{F} (A)}(L_{aa′}) = det_{End_{F} (A)}(L_{a}L_{a′}) = det_{End_{F} (A)}(L_{a}) det_{End_{F} (A)}(L_{a′}), where the last equality comes from Proposition 3.7(3) since End_{F} (A) is isomorphic to M_{m}(F). Note that

det_{End_{F} (A)}(L_{1_{A}}) = det_{M_{m}(F)}(1_{M_{m}(F)}) = 1

and det_{A}(1_{A}) = 1 by Proposition 5.3(1). Because F is infinite, Lemma 6.3 gives

det_{A}(aa′) = det_{A}(a) det_{A}(a′)

as desired.

To summarize: We defined the characteristic polynomial of an element in a finite-dimensional F -algebra in section 2. We defined the determinant det_{A} to be the constant term of this polynomial (Definition 3.8). In the case A = M_{n}(F), we found that det_{M_{n}(F)} is given by the Jacobi formula (Proposition 3.7(2)), hence the product formula holds for det_{M_{n}(F)} (Proposition 3.7(3)). Finally, we used Proposition 5.1 to prove the product formula for det_{A} (Theorem 6.1).

7 MISCELLANEOUS REMARKS.

This section is a survey of related results. It is necessarily briefer and more technical than the rest of the paper.

The “usual” definition of the characteristic polynomial.

For a in A, we claim that the formula

chpoly_{a} = det(x · 1_{A} − a)  (7.1)

holds in F [x]. But what does the expression “det(x · 1_{A} − a)” mean? For matrices, the determinant is given by the Jacobi formula involving the entries of a matrix, a formula that makes sense whether the entries are elements of F or polynomials in F [x]. The same reasoning holds for our more general notion of determinant.

One way to prove (7.1) is as follows. First prove it for the algebra A of upper-triangular matrices from Example 2.6. Then prove (7.1) for A = M_{n}(F) by reducing to the upper-triangular case as in the proof of Theorem 3.4. This implies (7.1) for general A by the arguments in [19, p. 225].

The characteristic polynomial of a product.

If a and b are n-by-n matrices, it is well known that the products ab and ba have the same characteristic polynomials. We can prove this statement directly for general A. Let c_{i} be any of the coefficients of the characteristic polynomial as in (2.10). Then c_{i}: A → F, and we want to show that c_{i}(ab) = c_{i}(ba) for all i and for all a and b in A. Fix i.

First assume that b is invertible. Then

c_{i}(ab) = c_{i}(b(ab)b^{−1}) = c_{i}(ba)

by Proposition 2.11(2). Now the set U of pairs (a, b) in A × A such that b is invertible is a nonempty open subset of A × A in the Zariski topology by Corollary 5.3. The map A × A → F defined by

(a, b) ↦ c_{i}(ab) − c_{i}(ba)

is given by a polynomial in the coordinates of a and b and is zero on U, hence is identically zero on A × A. This proves that c_{i}(ab) = c_{i}(ba) for all a and b in A. (This argument also works for A = M_{n}(F) with the traditional definition of the characteristic polynomial.)

Octonions, Jordan algebras, etc.

The results of sections 2 and 5 hold for many nonassociative algebras A. Specifically, suppose that A is an F -algebra in the sense of Definition 2.1, except that the multiplication need not be associative. (We still require that elements of F associate with every pair of elements of A, as in (2.2).) We say that A is power-associative if the value of a product aa ⋯ a does not depend on where one puts the parentheses. We say that A is strictly power-associative if K ⊗ A is power-associative for every field extension K of F. For example, the octonions are strictly power-associative. So is every Jordan algebra over a field of characteristic different from 2. When F is infinite, the algebra A is strictly power-associative if and only if it is power-associative.

The statements of section 2, Proposition 5.1, and parts (1) and (3) of Corollary 5.3 hold verbatim if we merely require that A be strictly power-associative and finite-dimensional. The proofs go through with essentially no change. In section 5, the left regular representation A → End_{F} (A) is just an injection of vector spaces. But the restriction to the subring generated by F and a does preserve multiplication for every a in A, and this is enough to prove Proposition 5.1 and parts (1) and (3) of Corollary 5.3. The statement of Proposition 5.3(2) has to be modified slightly (see [19, Corollary 2, p. 227] for a correct version).

The product formula (Theorem 6.1) does not hold in this level of generality. For example, it fails when A is a 27-dimensional exceptional Jordan algebra.^{4}

The Pfaffian.

Suppose that the characteristic of F is not 2. Write Skew_{n}(F) for the vector space of skew-symmetric n-by-n matrices, i.e., matrices a such that a^{t} = −a. If n is odd, then det(−a) = −det(a) by Proposition 2.11(3). Since det(a^{t}) = det(a) by Proposition 2.11(2), all n-by-n skew-symmetric matrices have determinant 0.

4These algebras correspond to Lie algebras of type F_{4} in the Killing-Cartan classification.

Their determinants are 27-dimensional cubic forms associated with Lie algebras of type E_{6}.

Suppose next that n is even. There is a polynomial map Pf : Skew_{n}(F) → F such that

(Pf(a))^{2} = det_{M_{n}(F)}(a)  (7.2)

called the Pfaffian (see, for example [22, chap. 15, sec. 9]). It seems a bit mysterious! (Note that equation (7.2) determines the Pfaffian only up to sign. Classically one chooses an invertible skew-symmetric matrix S with det(S) = 1 and fixes the sign of Pf so that Pf(S) = 1.)

In fact, the Pfaffian exists as a consequence of the characteristic polynomial as defined in section 2. Fix a matrix S as in the preceding paragraph and define a multiplication · on Skew_{n}(F) by

a · b = \frac{1}{2}(aS^{−1}b + bS^{−1}a).

This makes Skew_{n}(F) into a Jordan F -algebra with identity element S. These algebras arise naturally in the classification of central simple Jordan algebras.^{5} They are associated with nondegenerate skew-symmetric bilinear forms on F^{n}.

As described in the previous subsection, the theory developed in section 2 delivers a characteristic polynomial and a determinant for this Jordan algebra. One finds that the determinant is the Pfaffian:

det_{Skew_{n}(F)}(a) = Pf(a)

(see [19, pp. 230–232]).

Matrices over rings.

We have given a uniform construction of an analogue of the characteristic polynomial (hence the determinant) for every algebra over a field F. But the characteristic polynomial and determinant are typically defined for M_{n}(R), where R is merely a commutative ring with 1. Our definition also gives a characteristic polynomial in that case.

First, consider M_{n}(ℤ) as a ℤ-algebra. Since ℤ is a Noetherian unique factorization domain and M_{n}(ℤ) is a free ℤ-module of finite rank, the results in section 2 hold with A = M_{n}(ℤ) and F = ℤ. (In Proposition 2.11(5), we require that B be a direct summand of A.) The argument in Lemma 2.9 gives that the characteristic polynomial of a matrix a in M_{n}(ℤ) is the same as the characteristic polynomial of a considered as a matrix in M_{n}(ℚ), which is the usual characteristic polynomial by Theorem 3.4.

A coefficient c_{i} of the characteristic polynomial is given by an element of ℤ[t_{1}, t_{2}, …, t_{m}], and it is evaluated on a matrix a as described in Remark 6.2. For example, undergraduates are taught how to evaluate c_{n} = det, which is given by the Jacobi formula with respect to the standard basis of M_{n}(ℤ). In this manner, one can evaluate c_{i} on a matrix in R ⊗ M_{n}(ℤ) = M_{n}(R) for any commutative ring R with 1. This gives a characteristic polynomial—the usual one—for every a in M_{n}(R).

5See [19, chap. 5, sec. 7]. They correspond to Lie algebras of type C in the Killing-Cartan classification.

ACKNOWLEDGMENTS. I thank Adrian Wadsworth, Eric Brussel, and the referees for their comments, which greatly improved the structure and exposition of this paper.

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[28] K. Weierstrass, Zur Determinantentheorie, note based on lectures given in 1886/87; posthumously in Mathematische Werke, vol. 3, J. Knoblauch, ed., Mayer and Müller, Berlin, 1903, pp. 271–286.

Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322 http://www.mathcs.emory.edu/˜ skip skip@member.ams.org