
Karl Hessenberg
by JensPeter
Zemke, Institut für
Numerische Simulation, Technische
Universität HamburgHarburg, Hamburg,
Germany.
This page provides information on the German
mathematician and electrical engineer Karl
Hessenberg (19041959) 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. (DoktorIngenieur). 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 greatgrandfather 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 HamiltonCayleyschen
Gleichung, Karl Hessenberg, Dissertation,
1940 or 1941,
which can be translated as Solution of Linear
Eigenvalue Problems Using the HamiltonCayley
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
HamiltonCayleyschen 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 HamiltonCayley 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 HamiltonCayley 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
HamiltonCayleyschen 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 HamiltonCayley equation for the
solution of the secular equation and for the
Frobenius covariants, denoted here as
unitycomponent 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 nonstandard
analysis (1958).
Further Reading
Prof. Seiji Fujino was the first to be interested
in the life and work of Karl Hessenberg. Using
NADigest in his investigation, he managed to get
in touch with the widow and the heirs of Karl
Hessenberg. He has published several articles
containing more information:

Auf den Spuren eines deutschen
Wissenschaftlers; Dr. Karl Hessenberg, der von
der Geschichtsschreibung der Numerik vergessen
wurde (in German), Seiji Fujino, GAMM
Mitteilungen 18(2), 1995, 112114.

Who was Karl Hessenberg ?, Seiji
Fujino and Erhard Heil, INFORMATION 1(1), 1998,
2936.
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. 