Karl Hessenberg

by Jens-Peter Zemke, Institut für Numerische Simulation, Technische Universität Hamburg-Harburg, Hamburg, Germany.

This page provides information on the German mathematician and electrical engineer Karl Hessenberg (1904-1959) and the origin of the “Hessenberg matrix”.

Curriculum Vitae

Karl Adolf Hessenberg was born on September 8th, 1904 in Frankfurt am Main to Eduard Hessenberg, a lawyer, and his wife Emma, née Kugler. He went to school in Frankfurt am Main between 1911 and 1923, and then joined Hartmann & Braun AG as a trainee in Frankfurt am Main between 1923 and 1925. He then studied electrical engineering at the Technische Hochschule Darmstadt from 1925 to 1930. Between 1931 and 1932 he was an assistant of Prof. Alwin Walther at the Technische Hochschule Darmstadt. Afterwards, Hessenberg worked for the Elektrizitätswerk Rheinhessen in Worms. Starting in 1936 he was employed by A.E.G. as an engineer, first in Berlin, and later, until his untimely death, in Frankfurt am Main.

He finished his dissertation under the supervision of Prof. Alwin Walther at the Technische Hochschule Darmstadt with the grade Dr.-Ing. (Doktor-Ingenieur). It turned out that the dissertation contained some material already known, which delayed the submission to May 3rd, 1940, and the day of the oral examination, to July 23rd, 1940. The dissertation was approved February 11th, 1942.

Hessenberg was the brother of the composer Kurt Hessenberg, and his great-grandfather was the famous medical doctor and author of juvenile stories Heinrich Hoffmann. The mathematician Gerhard Hessenberg was a near relative.

More information about his life before 1936 can be found in the dissertation (see the link to the PDF version of the dissertation below) and in the following handwritten curriculum vitae (PDF, 184KB):

curriculum vitae of Karl Hessenberg (July 26th, 1936)

Karl Hessenberg died of cancer on February the 22nd, 1959.

The Origin of the Hessenberg Matrices

Most authors who cite Karl Hessenberg, including Lothar Collatz, Rudolf Zurmühl, Émile Durand, Alston Scott Householder, and James Hardy Wilkinson, cite the dissertation as the origin of the Hessenberg matrices with the title,

Auflösung linearer Eigenwertaufgaben mit Hilfe der Hamilton-Cayleyschen Gleichung, Karl Hessenberg, Dissertation, 1940 or 1941,

which can be translated as Solution of Linear Eigenvalue Problems Using the Hamilton-Cayley Equation. In truth, the title is:

Die Berechnung der Eigenwerte und Eigenlösungen linearer Gleichungssysteme, Karl Hessenberg, Dissertation, 1942 (5+5+175 pages, PDF, scan of a copy from Karl Hessenberg's estate, the first pages differ slightly from the published version, 28MB)

This can be translated as The Computation of the Eigenvalues and Eigensolutions of Linear Systems of Equations. The dissertation does not contain any Hessenberg matrices, even though a tridiagonal matrix is used as an example. On the second page of the preamble of the dissertation we can find the following comment,

„Es sei ferner bemerkt, dass auch das in Abschnitt V beschriebene Verfahren noch gewisser Verbesserungen fähig ist, worüber in einer besonderen Arbeit berichtet werden soll.”,

that is, “One can remark that the method described in Section V can be enhanced slightly, which will be reported in a separate paper.”. This separate paper is the first technical report (1. Bericht) of the IPM (Institute of Practical Mathematics) Darmstadt in the series „Numerische Verfahren” (“Numerical Methods”). It appeared on July, 23rd 1940, the day of Hessenberg's oral examination, and has the title:

Behandlung linearer Eigenwertaufgaben mit Hilfe der Hamilton-Cayleyschen Gleichung, Karl Hessenberg, 1. Bericht der Reihe Numerische Verfahren, 1940 (1+36 pages, PDF, scan of a copy from Karl Hessenberg's estate, 7MB),

that is, Treatment of Linear Eigenvalue Problems Using the Hamilton-Cayley Equation. In this paper Karl Hessenberg uses for the first time in history what we term nowadays “Hessenberg” matrices—see page 23, equation (58).

Hence, the wrong citations of the origin of the Hessenberg matrices might have arisen from the confusion between the dissertation and the report.

Yet, a probably more convincing hypothesis is that the original title of the dissertation was as stated in the citations, but that it would have been changed between the submission and the acceptance. It was recognized that Karl Hessenberg had found results already known and published in the 1936 book Elementary Matrices by Frazer, Duncan & Collar, namely the use of the Hamilton-Cayley equation and the Frobenius covariants to compute eigenvalues and eigenvectors. That this would have led to the delay in the acceptance of the dissertation is supported by the following statement:

„Der Verfasser der vorliegenden Dissertation hat verschiedene Dinge selbständig und unabhängig gefunden, von denen sich nachträglich herausgestellt hat, daß sie bereits bekannt waren. Das trifft z. B. zu für den Gebrauch der Hamilton-Cayleyschen Gleichung zur Auflösung der Säkulargleichung oder für die hier als Einheitsteilmatrizen bezeichneten Frobeniusschen Kovarianten. Wir haben es trotzdem für richtig gehalten, die Dissertation in der ursprünglichen Fassung endgültig einreichen zu lassen, während in dem geplanten Auszug in einer Zeitschrift diejenigen Kürzungen vorgenommen werden sollen, die durch Bezugnahme auf Literatur möglich sind.”

This statement was added March, 4th 1942 by his supervisor Alwin Walther and the second supervisor C. Schmieden on the second page of the published version of the Dissertation. A rough translation is given as follows:

“The author of the present dissertation, acting on his own initiative and independently, has found results that turn out to be already known. This is true, for example, for the use of the Hamilton-Cayley equation for the solution of the secular equation and for the Frobenius covariants, denoted here as unity-component matrices. We found it nevertheless adequate to submit the Dissertation in the original version, whereas the planned publication of extracts in a journal should be shortened whenever this is possible by using references to the literature.”

The second supervisor C. Schmieden most probably is Curt Schmieden, together with Detlef Laugwitz one of the founders of a kind of non-standard analysis (1958).

Acknowledgements

Brigitte Bossert, daughter of Karl Hessenberg, is gratefully acknowledged for this page and for making the PDF documents publicly available. Part of the text is based on the work of Prof. Seiji Fujino and Brigitte Bossert. The author also wishes to thank Erhard Heil of TU Darmstadt and Austin Dubrulle for their help.

Karl Hessenberg
(deutsch)

Karl Hessenberg
(english)

Karl Hessenberg

Curriculum Vitae

The Origin of the Hessenberg Matrices

Further Reading

Acknowledgements