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16. Loose Ends

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Abstract

As its title suggests, this chapter is devoted to tying up several historical loose ends related to the work of Frobenius featured in the previous chapters. The first section focuses on work done by Frobenius in response to the discovery of a gap in Weierstrass’ theory of elementary divisors as it applied to families of quadratic forms. Frobenius gave two solutions to the problem of filling the gap. The first drew upon the results and analogical reasoning used in his arithmetic theory of bilinear forms and its application to elementary divisor theory (Chapter 8).

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Vorheriges Kapitel Characters and Representations After 1897

Nächstes Kapitel Nonnegative Matrices

This fact and the other information preliminary to Frobenius’ discovery of a proof is drawn from Frobenius’ account in 1894 [203, pp. 577–578].

In the case of algebraic integers, some results require excluding p a prime divisor of 2 [203, p. 584] in the definition of “regular with respect to p.”

On Kronecker’s work with fields, see Purkert’s study [491].

Frobenius’ theorem actually specifies that \(\deg \chi = m - 1\), where m is the degree of the minimal polynomial of U. This additional information can be inferred from Frobenius’ proof; see below.

This is indeed correct, as Frobenius showed in his 1878 paper [181, p. 355], assuming that when \(f(t) = p(t)/q(t)\), detq(A) ≠ 0, so that [q(A)]^− 1 exists.

If L ² = M, then \({L}^{4} = {M}^{2} = 0\). This means that the characteristic roots of L must all be 0 and so (since L is 2 ×2) \(\varphi (t) = {t}^{2}\) is the characteristic polynomial of L. The Cayley–Hamilton theorem then implies \({L}^{2} = 0\not =M\), and so \(\sqrt{M}\) does not exist.

For the personal and institutional background to Sylvester’s brief flurry of interest in matrix algebra, see [462, pp. 135–138]. A fairly detailed mathematical discussion of Sylvester’s work on matrix algebra is given in [270, §6].

I refer to Lagrange’s attempt to extend his elegant generic solution to \(\ddot{y} + Ay = 0\), y(0) = y ₀, A n ×n, to the case in which \(f(\rho ) =\det {(\rho }^{2} + A)\) has one root of multiplicity two. See Section 4.2.1.

See in this connection Frobenius’ use of power and Laurent series in his proof of his minimal polynomial theorem (Theorem 7.2) and in his proof that a real orthogonal matrix can be diagonalized (Theorem 7.15).

Frobenius’ proof [208, pp. 697ff.] is expressed somewhat more generally than expounded here so as to allow a brief discussion of the problematic case det U = 0 as well as other functions of a matrix.

Frobenius proved that if S is a matrix with the property that the sequence S ^j, j = 0, 1, 2, …, has only a finite number of distinct terms, then all characteristic roots of S are either 0 or roots of unity, and the elementary divisors corresponding to the roots of unity are all linear [181, Satz VI, p. 357].

See in this connection [270, p. 107n.15].

Regarding Muth’s life and work, see [463].

In the preface, Muth wrote that he had been encouraged by the fact that “From the outset my undertaking was of special interest to several outstanding experts in the theory of elementary divisors,” namely Frobenius, S. Gundelfinger, and K. Hensel (Kronecker’s former student) [450, p. iv].

See, e.g., [25, pp. 297–301], [127, pp. 120–125], [567, pp. 130–131], [430, pp. 60–61], [240, v. 2, 41–42].

Kronecker ignored the work of Smith, even though Frobenius referenced it throughout his paper.

For a contemporary rendition, see, e.g., [141, pp. 459ff.].

Weber’s Lehrbuch der Algebra in its various editions from 1895 onward did not treat elementary divisor theory.

Bôcher also mentioned (1) λ-polynomials with integer coefficients and (2) polynomials in several variables; but it is unclear what he thought could be proved in these two cases, neither of which involves a principal ideal domain.

Let \(A ={\biggl ( \begin{array}{ccc} \rho _{1}\rho _{2}\! & \!0 \\ 0\! & \!1 \\ \end{array} \biggr )}\) and \(B ={\biggl ( \begin{array}{ccc} \rho _{1}\! & \!0 \\ 0\! & \!\rho _{2} \\ \end{array} \biggr )}\). Then A and B have the same invariant factors but B = P A Q is impossible, because it implies detPdetQ = 1 as well as that detP depends on the ρ _i and vanishes for \(\rho _{1} =\rho _{2} = 0\) [425, p. 107].

On the rationality group and Picard–Vessiot theory, see Gray’s historical account [255, pp. 267ff.].

Loewy refers to Schlesinger’s 1908 book on differential equations [514, pp. 157–158], which may have suggested to him the value of this connection for his own work, as described below.

The main results of Loewy’s theory are outlined in [427]. His most complete exposition is contained in a paper published in Mathematische Zeitschrift in 1920 and dedicated to Ludwig Stickelberger on the occasion of his 70th birthday [429]. The 1917 paper, however, brings out more fully the connections with Frobenius’ theory.

This result is implicit in [427] and explicit in [429, p. 102].

Krull’s companion matrices were defined as the negatives of Loewy’s because Krull defined the characteristic polynomial as det(t I − A), rather than as det(t I + A), as with Loewy.

These papers are [4]–[7] in Krull, Abhandlungen 1. For an overview of their contents, as well as Krull’s subsequent work on the theory of ideals, see P. Ribenboim’s essay on Krull [494, pp. 3ff.].

Despite his considerable mathematical talent, Remak, a Jew, did not have a correspondingly successful career as a mathematician and was eventually deported to Auschwitz, where he perished. For more on Remak’s life and work, see [440].

In the terminology and notation of Remak, \(\mathfrak{H}\) is the direct product \(\mathfrak{H} = \mathfrak{A} \times \mathfrak{B}\) of nontrivial subgroups \(\mathfrak{A}\), \(\mathfrak{B}\) if (1) every \(g \in \mathfrak{G}\) is expressible as g = a b, with \(a \in \mathfrak{A}\) and \(b \in \mathfrak{B}\); (2) \(\mathfrak{A} \cap \mathfrak{B} =\{ E\}\); (3) every \(a \in \mathfrak{A}\) commutes with every \(b \in \mathfrak{B}\). When \(\mathfrak{H}\) is abelian, this agrees with the definition of \(\mathfrak{H} = \mathfrak{A} \times \mathfrak{B}\) given by Frobenius and Stickelberger in 1878. Condition (3) must be added for nonabelian groups.

An automorphism σ of \(\mathfrak{H}\) is central if h ^− 1 σ(h) is in the center of \(\mathfrak{H}\) for all \(h \in \mathfrak{H}\) [493, p. 293], a notion still in use today.

The ascending chain condition for ideals had been introduced by Noether in her above-mentioned fundamental paper of 1921 [455, Satz I, p. 30] and goes back to Dedekind, as she noted. All Krull’s early papers on ideals cite this paper as a basic reference. Nine months before submitting his paper [375] on generalized abelian groups, Krull had submitted a paper with a descending chain condition on the successive powers of an ideal [374, p. 179, (f)]. As for Noether, her important work on what are now called Dedekind rings [456, 457], involved a descending chain condition. A bit later, Artin used a descending chain condition in his study of what are now called Artinian rings.

Krull spoke simply of indecomposable subgroups, but he meant the analogue of what Remak had called “directly indecomposable” subgroups.

See [375, p. 191 n.34]. Krull was referring to Loewy’s results on matrix complexes (described above) as they apply to the completely reducible operators of Loewy’s Theorem 16.8. Of course, it should be kept in mind that although Krull’s theorem applies to any \(\mathfrak{A}\), the \(\mathfrak{D}_{ii}\) are simply directly indecomposable and not necessarily irreducible in Loewy’s sense.

Krull did not use the terminology of linear transformations and vector spaces, which was soon brought into linear algebra primarily through the influence of the work of Weyl (see below).

Krull’s matrix of Θ with respect to a basis is the transpose of the usual definition. Thus he gets the transpose of C _i [376, p. 26], which agrees with how companion matrices are defined in his doctoral dissertation [372, p. 58].

Lasker reserved the term “ideal” for an ideal in Z[x ₁, …, x _n].

Châtelet also extended his notion of module to “smireal” n-dimensional spaces [88, pp. 10, 25].

In §§106–107, van der Waerden also allowed \(\mathfrak{R}\) to be what one might call a “noncommutative Euclidean domain” [568, p. 120].

That is, detU, which belongs to \(\mathfrak{R}\), has an inverse in \(\mathfrak{R}\). Thus \({U}^{-1} = {(\det U)}^{-1}\mathrm{Adj}\,U\) is a matrix over \(\mathfrak{R}\), and \(x = u{U}^{-1}\).

Sperner explained in the preface that the subject matter of the book was as Schreier intended, although the ordering of the material and some proofs were changed in part to achieve greater simplicity.

In Space–Time–Matter, first published in 1921, Weyl developed the machinery of tensor algebra within the context of abstract vector spaces. In his 1923 monograph Mathematical Analysis of the Space Problem [601, Anhang 12], Weyl developed elementary divisor theory over \(\mathbb{C}\) within the same context and so made the concept of a linear transformation acting on a vector space fundamental. For more on Weyl and the space problem, see [518, §2.8] and [276, §11.2].

For a discussion of Schreier’s important work on continuous groups, see [276, pp. 497ff.].

Under Schmidt’s influence, van der Waerden devoted an entire chapter [568, Ch. 6] to groups with operators [569, p. 34].

As with Theorem 16.11, \(\mathfrak{R}\) can also be a “noncommutative Euclidean domain.” See the citation at that theorem.

25.

M. Bôcher. Introduction to Higher Algebra. Macmillan, New York, 1907. Republished by Dover Publications, New York, 1964. German translation as [26].

26.

M. Bôcher. Einführung in die höhere Algebra. Teubner, Leipzig, 1910. Translated by H. Beck.

84.

A. Cayley. A memoir on the theory of matrices. Phil. Trans. R. Soc. London, 148:17–37, 1858. Reprinted in Papers 2, 475–496.

88.

A. Châtelet. Leçons sur la théorie des nombres. (Modules. Entiers algébrique. Réduction continuelle.). Gauthier–Villars, Paris, 1913.

127.

L. E. Dickson. Modern Algebraic Theories. Sandborn, Chicago, 1926.MATH

141.

D. Dummit and R. Foote. Abstract Algebra. Wiley, 2nd edition, 1999.

175.

G. Frobenius. Über den Begriff der Irreductibilität in der Theorie der linearen Differentialgleichungen. Jl. für die reine u. angew. Math., 76:236–270, 1873. Reprinted in Abhandlungen 1, 106–140.

181.

G. Frobenius. Über lineare Substitutionen und bilineare Formen. Jl. für die reine u. angew. Math., 84:1–63, 1878. Reprinted in Abhandlungen 1, 343–405.

182.

G. Frobenius. Theorie der linearen Formen mit ganzen Coefficienten. Jl. für die reine u. angew. Math., 86:146–208, 1879. Reprinted in Abhandlungen 1, 482–544.

185.

G. Frobenius. Theorie der linearen Formen mit ganzen Coefficienten (Forts.). Jl. für die reine u. angew. Math., 88:96–116, 1880. Reprinted in Abhandlungen 1, 591–611.

203.

G. Frobenius. Über die Elementartheiler der Determinanten. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 7–20, 1894. Reprinted in Abhandlungen 2, 577–590.

208.

G. Frobenius. Über die cogredienten Transformationen der bilinearen Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 7–16, 1896. Reprinted in Abhandlungen 2, 695–704.

209.

G. Frobenius. Über vertauschbare Matrizen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 601–614, 1896. Reprinted in Abhandlungen 2, 705–718.

234.

G. Frobenius and I. Schur. Über die äquivalenz der Gruppen linearer Substitutionen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 209–217, 1906. Reprinted in Frobenius, Abhandlungen 3, 378–386.

240.

F. Gantmacher. Matrix Theory. AMS Chelsea Publishing, 2000. This work, published in two volumes, is an English translation of Gantmacher’s Teoriya Matrits (Moscow, 1953). It first appeared in 1959.

255.

J. Gray. Linear Differential Equations and Group Theory from Riemann to Poincaré. Birkhäuser, Boston, 2nd edition, 2000.MATH

264.

O. Haupt. Einführung in die Algebra, volume 2. Akademische Verlaggesellschaft, Leipzig, 1929. The theory of generalized abelian groups and its application to elementary divisor theory is presented by Krull in an appendix (pp. 617–629).

270.

T. Hawkins. Another look at Cayley and the theory of matrices. Archives internationales d’histoire des sciences, 26:82–112, 1977.MathSciNet

276.

T. Hawkins. Emergence of the Theory of Lie Groups. An Essay on the History of Mathematics 1869–1926. Springer, New York, 2000.

280.

L. Heffter. Einleitung in die Theorie der linearen Differentialgleichungen mit einer unabhängigen Variable. B. G. Teubner, Leipzig, 1894.

282.

K. Hensel. Ueber die Elementheiler componirter Systeme. Jl. für die reine u. angew. Math., 114:109–115, 1895.

283.

K. Hentzelt and E. Noether. Bearbeitung von K. Hentzelt: zur Theorie der Polynomideale und Resultanten. Math. Ann., 88:53–79, 1923. Reprinted in E. Noether, Abhandlungen, 409–435.

328.

C. Jordan. Mémoire sur les équations différentielles linéaires à intégrale algébrique. Jl. für die reine u. angew. Math., 84:89–215, 1878. Reprinted in Oeuvres 2, 13–139.

350.

A. Krazer. Lehrbuch der Thetafunktionen. Teubner, Leipzig, 1903. Reprinted by Chelsea Publishing Company (New York, 1970).

359.

L. Kronecker. Über die congruenten Transformation der bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 397–447, 1874. Presented April 23, 1874. Reprinted in Werke 1, 423–483.

366.

L. Kronecker. Über die Composition der Systeme von n ² Grössen mit sich selbst. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 1081–1088, 1890. Reprinted in Werke 3₁, 463–473.

367.

L. Kronecker. Algebraische Reduction der Schaaren bilinearer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 1225–1237, 1890. Reprinted in Werke 3₂, 141–155.

368.

L. Kronecker. Algebraische Reduction der Schaaren quadratischer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 1375–1388, 1890. Reprinted in Werke 3₂, 159–174.

369.

L. Kronecker. Algebraische Reduction der Schaaren quadratischer Formen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 9–17, 34–44, 1891. Reprinted in Werke 3₂, 175–198.

370.

L. Kronecker. Reduction der Systeme von n ² ganzahlige Elementen. Jl. für die reine u. angew. Math., 107:135–136, 1891. Reprinted in Werke 4, 123–124.

372.

W. Krull. Über Begleitmatrizen und Elementartheiler. Inauguraldissertation Universität Freiburg i. Br., 1921. First published in Krull’s Abhandlungen 1 (1999), 55–95.

373.

W. Krull. Algebraische Theorie der Ringe. I. Math. Ann., 88:80–122, 1922. Reprinted in Abhandlungen 1, 80–122.

374.

W. Krull. Algebraische Theorie der Ringe. II. Math. Ann., 91:1–46, 1924. Reprinted in Abhandlungen 1, 166–211.

375.

W. Krull. Über verallgemeinerte endliche Abelsche Gruppen. Mathematische Zeitschrift, 23:161–196, 1925. Reprinted in Papers 1, 263–298.

376.

W. Krull. Theorie und Anwendung der verallgemeinten Abelschen Gruppen. S’ber. Akad. Wiss. Heidelberg, Math.-Natur. Kl., 1:1–10, 1926. Reprinted in Papers 1, 299–328.

394.

E. Landau. Ein Satz über die Zerlegung homogener linearer Differentialausdrücke in irreducible Factoren. Jl. für die reine u. angew. Math., 124, 1902.

418.

R. Lipschitz. Beweis eines Satzes aus der Theorie der Substitutionen. Acta Mathematica, 10:137–144, 1887.MathSciNetMATHCrossRef

421.

A. Loewy. Ueber die irreduciblen Factoren eines linearen homogenen Differentialausdruckes. Berichte über d. Verh. d. Sächsischen Gesell. der Wiss., math. -phys. Klasse, pages 1–13, 1902.

422.

A. Loewy. Über die Reducibilität der Gruppen linearer homogener Substitutionen. Trans. American Math. Soc., 4:44–64, 1903.MathSciNetMATH

424.

A. Loewy. Über reduzible lineare homogene Differentialgleichungen. Math. Ann., 56:549–584, 1903.MathSciNetMATHCrossRef

425.

A. Loewy. Kombinatorik, Determinanten und Matrices. In P. Epstein and H. E. Timmerding, editors, Repertorium der höheren Mathematik, volume 1, chapter 2. Leipzig and Berlin, 1910.

426.

A. Loewy. Über lineare homogene Differentialsysteme und ihre Sequenten. Sitzungsberichte der Heidelberger Akademie der Wissenschaften, math.–naturwis. Kl., Abt. A, Abhandlung 17, 1913.

427.

A. Loewy. Die Begleitmatrix eines linearen homogenen Differentialausdruckes. Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augustus-Universität zu Göttingen, pages 255–263, 1917.

428.

A. Loewy. Über Matrizen- und Differentialkomplexe. Math. Ann., 78:1–51, 1917.MathSciNetMATHCrossRef

429.

A. Loewy. Begleitmatrizen und lineare homogene Differentialausdrücke. Mathematische Zeitschrift, 7:58–125, 1920.MathSciNetMATHCrossRef

430.

C. C. MacDuffee. The Theory of Matrices. Springer, Berlin, 1933.CrossRef

440.

U. Merzbach. Robert Remak and the estimation of units and regulators. In S. S. Demidov et al, editor, Amphora: Festschrift für Hans Wußing zu seinem 65. Geburtstag, pages 481–552. Birkhäuser, Berlin, 1992.

450.

P. Muth. Theorie und Anwendung der Elementartheiler. Teubner, Leipzig, 1899.

455.

E. Noether. Idealtheorie in Ringbereichen. Math. Ann., 83:24–66, 1921. Reprinted in Abhandlungen, 354–366.

456.

E. Noether. Abstrakter Aufbau der Idealtheorie im algebraischen Zahlkörper. Jahresbericht der Deutschen Mathematiker-Vereinigung, 33:102, 1924. Reprinted in Abhandlungen, p. 102.

457.

E. Noether. Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern. Math. Ann., 96:26–61, 1926–27. Reprinted in Abhandlungen, 493–528.

458.

E. Noether. Hypercomplexe Grössen und Darstellungstheorie. Mathematische Zeitschrift, 30:641–692, 1929. Reprinted in Abhandlungen, 563–614.

459.

E. Noether and W. Schmeidler. Moduln in nichtkommutativen Bereichen, insbesondere aus Diffzenzausdrücken. Mathematische Zeitschrift, 8:1–35, 1920. Reprinted in Abhandlungen, 318–352.

462.

K. H. Parshall and D. Rowe. The Emergence of the American Mathematical Research Community, 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore. History of Mathematics, Vol. 8. American Mathematical Society, 1994.

463.

M. Pasch. Peter Muth. Jahresbericht der Deutschen Mathematiker-Vereinigung, 18:454–456, 1909.

491.

W. Purkert. Zur Genesis des abstrakten Körperbegriffs. 1. Teil. NTM-Schriftenreihe für Geschichte der Naturwiss., Technik, und Med., 8:23–37, 1971.

493.

R. Remak. Über die Zerlegung der endlichen Gruppen in direkte unzerlegbare Faktoren. Jl. für die reine u. angew. Math., 139:293–308, 1911.MATH

494.

P. Ribenboim. Wolfgang Krull—Life, Work and Influence. In Wolfgang Krull Gesammelte Abhandlingen, volume 1, pages 1–20. Walter de Gruyter, Berlin, 1999.

513.

L. Schlesinger. Handbuch der Theorie der linearen Differentialgleichungen, volume 1. B. G. Teubner, 1895.MATH

514.

L. Schlesinger. Vorlesungen über Differentialgleichungen. B. G. Teubner, Leipzig, 1908.

515.

O. Schmidt. Über unendliche Gruppen mit endlicher Kette. Jl. für die reine u. angew. Math., 29:34–41, 1928–1929.

518.

E. Scholz. Historical aspects of Weyl’s Raum–Zeit–Materie. In E. Scholz, editor, Hermann Weyl’s Raum–Zeit–Materie and a General Introduction to His Scientific Work. Birkhäuser, 2000.

520.

O. Schreier and E. Sperner. Vorlesungen über Matrizen. B. G. Teubner, Leipzig, 1932.

550.

L. Stickelberger. Ueber Schaaren von bilinearen und quadratischen Formen. Jl. für die reine u. angew. Math., 86:20–43, 1879.

559.

J. J. Sylvester. On the equation to the secular inequalities in the planetary theory. Phil. Mag., 16:110–11, 1883. Reprinted in Papers, v. 4, 110–111.

567.

H. W. Turnbull and A. C. Aitken. An Introduction to the Theory of Canonical Matrices. Blackie and Son, London & Glasgow, 1932.

568.

B. L. van der Waerden. Moderne Algebra, volume 2. Springer, Berlin, 1931.

569.

B. L. van der Waerden. On the sources of my book Moderne Algebra. Historia Mathematica, 2:31–40, 1975.

585.

J. H. M. Wedderburn. Lectures on Matrices. American Mathematical Society, New York, 1934.

588.

K. Weierstrass. Zur Theorie des quadratischen und bilinearen Formen. Monatsberichte der Akademie der Wiss. zu Berlin, pages 311–338, 1868. Reprinted with modifications in Werke 2, 19–44.

601.

H. Weyl. Mathematische Analyse des Raumproblems. Springer, Berlin, 1923.CrossRef

Titel: Loose Ends
verfasst von: Thomas Hawkins
Verlag: Springer New York
Buch: The Mathematics of Frobenius in Context
Print ISBN: 978-1-4614-6332-0

Electronic ISBN: 978-1-4614-6333-7

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-6333-7_16

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