Abstract
From the time of his dissertation (1870) through his work on the problem of Pfaff in 1876 (Chapter 6), Frobenius’ penchant for working on algebraic problems had been pursued within the framework of analysis, especially differential equations. As we saw, his work on the problem of Pfaff—ostensibly a problem within the field of differential equations—had engaged him more fully with linear algebra.
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Notes
- 1.
More detailed information about Dedekind can be found at the beginning of Chapter 2.
- 2.
Clark’s translation has “transcendental Arithmetic,” but “higher Arithmetic” is closer to the original Latin.
- 3.
Recall that \({P}^{-1} = {(\det P)}^{-1}\mathrm{Adj}\,P\), where Adj P is the transposed matrix of cofactors of P. Thus detP = ± 1 means that P − 1 = ± Adj P is integral.
- 4.
Let \(a_{1},\ldots,a_{m}\) denote the (1 ×n) rows of A and \(b_{1},\ldots,b_{m}\) the (1 ×n) columns of B. Then A B = I m means that \(a_{i}b_{j} =\delta _{ij}\). Thus for \({A}^{{\ast}}{B}^{{\ast}} = I_{m+1}\) to hold, it is necessary that \(ab_{j} = 0\) for all j, i.e., that \(aB = 0\). Since m < n, we can certainly determine an integral \(a\not =0\) such that \(aB = 0\). Furthermore, we can take \(a\) such that \(\gcd a = 1\). It then follows that an integral column matrix \(b\) may be determined such that \(ab =\sum a_{j}b_{j} = 1\), where \(b\) is the n ×1 column matrix with the b j as entries. Thus the (m + 1, m + 1) entry of A ∗ B ∗ will be 1, as required. However, for k < m + 1, the (m + 1, k) entries must all be zero, i.e., we must have \(a_{i}b = 0\) for all j, or equivalently, \(Ab = 0\). The chosen \(b\) need not satisfy this condition, and so we replace \(b\) with \(b - c\). To have \(a(b - c) = 1\),e we need that \(ac = 0\). To this end, we may choose \(c = Bx\) for any \(x\), for then \(ac = a(Bx) = (aB)x = 0x = 0\). We also need that \(A(b - c) = 0\). But \(A(b - c) = Ab - ABx = Ab - x\), since \(AB = I_{m}\). Thus we need to set \(x = Ab\), and so \(c = BAb\). The desired matrices are \({A}^{{\ast}} ={\biggl ( \begin{array}{c} A \\ a \end{array} \biggr )}\) and \({B}^{{\ast}} = \left (\begin{array}{cc} B &c - BAc\\ \end{array} \right )\):
$$\displaystyle{{A}^{{\ast}}{B}^{{\ast}} = \left (\begin{array}{cc} AB &Ab - (AB)Ab \\ aB & ab - (aB)Ab\\ \end{array} \right ) = \left (\begin{array}{cc} I_{m}&0 \\ 0 & 1\\ \end{array} \right ),}$$since A B = I m by hypothesis and \(aB = 0\) by construction.
- 5.
If p t is the given row vector of Hermite’s lemma, since \(\gcd {p}^{t} = 1\), integers \(q_{1},\ldots,q_{n}\) may be determined such that \(\sum _{i=1}^{n}p_{i}q_{i} = 1\). Then if \(q\) denotes the column vector with entries q i , \({p}^{t}q = 1\), and Frobenius’ corollary applies and yields Hermite’s lemma. To show that Hermite’s lemma implies Frobenius’ corollary, observe that \({p}^{t}q = 1\) implies \(\gcd p =\gcd q = 1\).
- 6.
For example, if \(F(x) = 2x_{1}^{2} + 6x_{1}x_{2} + 7x_{2}^{2}\), so that \(f =\gcd A = 1\), Gauss’ Theorem I [244, Art. 229] implies that \(F(x) = 1\) is impossible, since \(F(0, 1) = 7 \equiv 3\,(\mathrm{mod}\,4)\), and so all odd numbers representable by F must be congruent to 3 mod 4.
- 7.
If A 1 = 0, there is nothing to prove. If A 1 ≠ 0, i.e., if r = rank A > 1, then the reduction lemma may be applied to A 1 with \(f_{2} =\gcd A_{1}\). The conclusion is that unimodular P ′, Q ′ can be determined such that \(P^{\prime}A_{1}Q^{\prime} ={\biggl ( \begin{array}{cc} f_{2}\! & \!0 \\ \ 0\! & \!f_{2}A_{2} \end{array} \biggr )}\). Then
$$\displaystyle{P_{2} = \left (\begin{array}{cc} 1 & 0\\ 0 & P^{\prime}\\ \end{array} \right )\quad \mathrm{and}\quad Q_{2} = \left (\begin{array}{cc} 1 & 0\\ 0 & Q^{\prime}\\ \end{array} \right )}$$are unimodular, and block multiplication gives
$$\displaystyle{P_{2}(P_{1}AQ_{1})Q_{2} = \left (\begin{array}{cc} f_{1} & 0 \\ 0 & f_{1}P^{\prime}A_{1}Q^{\prime}\\ \end{array} \right ) = \left (\begin{array}{cccc} f_{1} & 0 & 0 \\ 0 & f_{1}f_{2} & 0 \\ 0 & 0 & f_{1}f_{2}A_{2}\\ \end{array} \right ).}$$If A 2 ≠ 0, i.e., if r > 2, we may apply the reduction lemma to A 2, and so on, until we end up with unimodular matrices \(P_{i},Q_{i}\), i = 1, …, r, such that P A Q, with \(P = P_{r}\cdots P_{1}\) and \(Q = Q_{1}\cdots Q_{r}\), has the form in (8.4). In other words, the reduction theorem follows from the reduction lemma.
- 8.
See Section 9.1.1 for the definition of form composition.
- 9.
Frobenius did not use this mode of expression, although he began using it a few years later as a result of work on a problem involving abelian functions. See Section 10.6.
- 10.
C = P A Q implies that every coefficient of C is an integral linear combination of coefficients of A; and \(A = {P}^{-1}C{Q}^{-1}\) implies that every coefficient of A is an integral linear combination of coefficients of C.
- 11.
This is the formula given verbally in [182, III, p. 513] in the special case K = k I m .
- 12.
- 13.
- 14.
Frobenius, usually an excellent expositor, failed to inform the reader of this easy consequence of his normal form theorem at the beginning of his 1880 paper. It was only after the difficult task of proving the necessity of the above condition was completed that its sufficiency was quickly deduced as above (but without matrix notation).
- 15.
For example, Frobenius never cited Cayley’s 1858 “A Memoir on Matrices,” which was also published in the Philosophical Transactions, even though he did cite Cayley’s earlier work, in Crelle’s Journal, on the Cayley–Hermite problem (Chapter 7).
- 16.
Smith was familiar with Cayley’s 1858 memoir on matrices [84] (discussed in Section 7.4); he referred to it in his report on the theory of numbers for the British Association for the Advancement of Science [538, p. 167].
- 17.
Recall from Section 6.3 that the determinant-theoretic definition of rank was first introduced by Frobenius in 1877, although the rank property was used earlier.
- 18.
That is say that \(x_{1},x_{2} \in {\mathbb{Z}}^{n}\) are equivalent if \(Ax_{1} \equiv Ax_{2}\,(\mathrm{mod}\,k)\). This is an equivalence relation. The number of incongruent \(x\) in an equivalence class equals the number P of solutions to \(Ax \equiv 0\,(\mathrm{mod}\,k)\), which is given by Smith’s Theorem 8.19, so if Q is the number of equivalence classes, then P Q = k n (the number of elements in \({(\mathbb{Z}/k\mathbb{Z})}^{n}\)) and Q equals the number of incongruent \(b\) such that (8.22) has a solution.
- 19.
Granted Smith’s lemma, if f = gcd(A), apply Smith’s lemma to A ′ = (1 ∕ f)A to obtain \(y\) such that \(u = A^{\prime}y\) and \(\gcd (u) = 1\). Thus \(fu = Ay\) and \(v = fu\) has \(\gcd (v) = f\), so that integers \(x_{1},\ldots,x_{m}\) exist for which \(\sum _{I=1}^{m}x_{i}w_{i} = f\). Thus if \(x ={ \left (\begin{array}{ccc} x_{1} & \cdots &x_{m}\\ \end{array} \right )}^{t}\), we have \({x}^{t}Ay = {y}^{t}w = f\), which is Frobenius’ conclusion. Conversely, if Frobenius’ Lemma 8.5 is assumed and gcd(A) = 1, then \({x}^{t}Ay = 1\). This says that if \(u = Ay\), then \({x}^{t}u = 1\), which means that \(\gcd (u) = 1\).
- 20.
- 21.
The set of parameters \((t_{1},\ldots,t_{d})\) such that these determinants do not vanish is open and dense in \({\mathbb{C}}^{d}\), but Frobenius did not indicate rational operations to determine such a choice of \((t_{1},\ldots,t_{d})\).
- 22.
If \(a(\lambda ) =\sum _{ i=0}^{\alpha }a{_{i}\lambda }^{i},b(\lambda ) =\sum _{ i=0}^{\beta }b{_{i}\lambda }^{i} \in \mathbb{F}[\lambda ]\) have degrees α and β ≤ α, then one wishes to determine the “quotient” \(q(\lambda ) =\sum _{ i=0}^{\alpha -\beta }c{_{i}\lambda }^{i}\) such that the “remainder” r(λ) = a(λ) − b(λ)q(λ) has degree less than β. If a(λ) − b(λ)q(λ) is expanded in powers of λ, then the coefficients c i of q(λ) must be determined so that the coefficients of λ i in the expansion vanish for all i ≥ β. This yields a system of α − β + 1 equations in the α − β + 1 unknown coefficients c i of q(λ) that since b β ≠ 0, yield, successively, unique values of \(c_{\alpha -\beta },c_{\alpha -\beta -1},\ldots,c_{0}\). For example, setting the coefficient of λ α equal to zero yields the equation \(a_{\alpha } - b_{\beta }c_{\alpha -\beta } = 0\), so \(c_{\alpha -\beta } = a_{\alpha }/b_{\beta }\).
- 23.
In his Traité des substitutions of 1870 [322, Livre I,§III], Jordan defined these fields as follows. Let f(x) be an irreducible polynomial of degree ν > 1 over the integers mod p. (He proved that such polynomials always exist by proving that \({x}^{{p}^{\nu } } - x\) always has irreducible factors of degree ν over the integers mod p.) Then “nothing prevents us” from introducing the “imaginary symbol” i, which is subject to the condition that \(f(i) \equiv 0\,(\mathrm{mod}\,p)\). (This i is not to be confused with \(\sqrt{-1}\).) Then for any polynomial F(x) with integer coefficients, \(F(x) \equiv f(x)q(x) + r(x)\,(\mathrm{mod}\,p)\), where degr(x) < ν. Since \(f(i) \equiv 0\,(\mathrm{mod}\,p)\), we have \(F(i) \equiv r(i)\,(\mathrm{mod}\,p)\). The remainder expressions r(i) are called “complex integers,” and they can be partitioned into p ν congruence classes mod p, which are the elements of GF(p ν). (In effect Jordan’s complex integers are the elements of \(\mathbb{F}_{p}[x]/(f(x))\) with i taken as the equivalence class of x.) Jordan also showed that \(i,{i}^{p},\ldots,{i}^{{p}^{\nu -1} }\) are the ν (distinct) roots of \(f(x) \equiv 0\,(\mathrm{mod}\,p)\). Of course, \({i}^{{p}^{k} }\) is congruent mod p to a complex integer, i.e., is an element of GF(p ν).
- 24.
The notation \(F[\varphi ]\) is mine, not Frobenius’. The matrix \(F[\varphi ]\) is related to what A. Loewy later called the companion matrix (Begleitmatrix) of \(\varphi\), as we will see in Section 16.3.1. I will refer to \(F[\varphi ]\) as the Frobenius companion matrix of \(\varphi\). As I have already explained, Frobenius himself never spoke of fields \(\mathbb{F}\) in general. In introducing \(F[\varphi ]\), he assumed for the sake of concreteness that the coefficients of \(\varphi\) are algebraic numbers “in a certain field” [182, p. 542].
- 25.
It should be noted than every nonsingular family \(\lambda A_{1} + A_{2}\) is equivalent to one in the form λ I + B, since \(A_{1}^{-1}(\lambda A_{1} + A_{2})I =\lambda I + B\), \(B = A_{1}^{-1}A_{2}\); and so in discussing invariant factors, only families λ I + B need be considered.
- 26.
The (α, 1) minor of \(F[\varphi ]\) is ( − 1)α, making d α − 1(r) = 1 and \(e_{\alpha }(r) = d_{\alpha }(r) =\varphi (r)\), while e i (λ) = 1, for all i < α.
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Hawkins, T. (2013). Arithmetic Investigations: Linear Algebra. In: The Mathematics of Frobenius in Context. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6333-7_8
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