# Semyon Aranovich Gershgorin

### Quick Info

Born
24 August 1901
Pruzhany, Russian Empire (now Belarus)
Died
30 May 1933
Leningrad, USSR (now St Petersburg, Russia)

Summary
Semyon Gershgorin was a Belarussian mathematician who produced important results in numerical approximation.

### Biography

Semyon Aranovich Gershgorin's name appears in various different transliterations in addition to the one we have chosen to use. His first name is often written as Semën or Semen, his patronymic is sometimes written as Aranovič, Aronivič or Aronovich, while his family name appears as Geršagorin, Gerschgorin or Gerszgorin. The family name is originally Yiddish and, somewhat confusingly, transliterations from the Yiddish can be given in the form Hirshhorn or Hirschhorn.

He studied at Petrograd Technological Institute from 1923, and defended an outstanding thesis submitted to the Division of Mechanics. The papers he published at this time are (all in Russian): Instrument for the integration of the Laplace equation (1925); On a method of integration of ordinary differential equations (1925); On the description of an instrument for the integration of the Laplace equation (1926); and On mechanisms for the construction of functions of a complex variable (1926). He became Professor at the Institute of Mechanical Engineering in Leningrad in 1930, and from 1930 he worked in the Leningrad Mechanical Engineering Institute on algebra, theory of functions of complex variables, numerical methods and differential equations. He became head of the Division of Mechanics at the Turbine Institute as well as teaching innovative courses at Leningrad State University. The authors of [5] write:-
... he quickly became one of the leading figures in Soviet Mechanics and Applied Mathematics. His many papers brought him to attention and made him famous world-wide. These were on the theory of elasticity, the theory of vibrations, the theory of mechanisms, methods of approximate numerical integration of differential equations and on other areas of mechanics and applied mathematics. His first publications showed him to be an outstandingly gifted young scientist but soon, in what were his last years, his talent matured and blossomed. The main features of Gershgorin's individuality are the innovative way he approached a problem, combined with power and clarity of analysis. These features are already apparent in his early papers - for example, his brilliant idea showing how to construct the profiles of airplane wings - and continue through to his last outstanding results in elasticity theory and in theory of vibrations.
First let us look briefly at his 1925 and 1926 papers whose titles we gave above. Garry Tee, in [6], describes some of Gershgorin's contributions contained in them:-
Gershgorin proposed an original and intricate mechanism for solving the Laplace equation, and he described such a device in detail in 'Instrument for the integration of the Laplace equation' (1925). J J Sylvester had proved that any algebraic relation between real variables could be modelled by linkage mechanisms, but he had not mentioned the possibility of actually constructing such mechanisms. In Gershgorin's 1926 paper 'On mechanisms for the construction of functions of a complex variable', he described linkage mechanisms implementing the complex arithmetic operations of addition, subtraction, multiplication and division. He described mechanisms for constructing the complex relations $w = z^{2}$ and $w = z^{3}$, which could also be applied for extracting square roots and cube roots. Gershgorin proposed that linkage mechanisms be constructed for various standard functions, which could then be assembled into larger mechanisms for more complicated functions. Later he became the first person to construct analogue devices applying complex variables to the theory of mechanisms ...
In 1929 Gershgorin published On electrical nets for approximate solution of the differential equation of Laplace (Russian) in which he gave a method for finding approximate solutions to partial differential equations by constructing a model based on networks of electrical components. In the following year he published Fehlerabschätzung für das Differenzverfahren zur Lösung partieller Differentialgleichungen in which he made a careful analysis of the convergence of finite-difference approximation methods for solving the Laplace equation. However, his most famous paper, published in 1931, is Über die Abgrenzung der Eigenwerte einer Matrix . In this paper he gave powerful estimates for matrix eigenvalues, known as his Circle Theorem. Richard Varga writes in [1]:-
The Gershgorin Circle Theorem , a very well-known result in linear algebra today, stems from the paper of S Gershgorin in 1931 where, given an arbitrary $n \times n$ >complex matrix, easy arithmetic operations on the entries of the matrix produce $n$ disks, in the complex plane, whose union contains all eigenvalues of the given matrix. The beauty and simplicity of Gershgorin's Theorem has undoubtedly inspired further research in this area, resulting in hundreds of papers in which the name "Gershgorin" appears.
Ludwig Elsner [2] makes an interesting remark concerning the fact that Gershgorin wrote his paper on the Circle Theorem in German:-
I would also point out the significance of the fact that Gershgorin's article was written in German. It is safe to guess that this was one of the reasons that certain people outside the Soviet Union, such as Olga Taussky-Todd and Alfred Brauer, studied it and made it known in the West during the 1940s. Other important papers written in the Soviet Union in the 1930s became known in the West only much later. I think that the influence of language on communication of mathematical ideas and results is substantial and cannot be overstated.
The authors of [5] write about Gershgorin's death at the very young age of 31:-
A vigorous, stressful job weakened Semyon Aranovich's health; he succumbed to an accidental illness, and a brilliant and successful young life has ended abruptly. Semyon Aranovich Gershgorin's death is a great and irreplaceable loss to Soviet Science.
Gershgorin's final paper On the conformal map of a simply connected domain onto a circle (Russian) was published in 1933 after his death. He begins the paper with the following sentence:-
In this paper we present a method of conformal mapping of a given (finite or infinite) connected domain onto a disk, which is based on reducing the problem to a Fredholm integral equation.
Garry Tee writes [6]:-
L Lichtenstein [in 'Zur Theorie der konformen Abbildung: Konforme Abbildung nicht-analytischer, singularitätenfreier Flächenstücke auf ebene Gebiete' (1916)] had reduced that important problem to the solution of a Fredholm integral equation. Independently of Lichtenstein, Gershgorin utilised Nystrom's method and reduced that conformal transformation problem to the same Fredholm integral equation. Later, A M Banin solved the Lichtenstein-Gershgorin integral equation approximately, by reducing it to a finite system of linear differential equations.

### References (show)

1. R S Varga, Gersgorin and His Circles (Springer-Verlag, Berlin, 2004).
2. L Elsner, Review: Gersgorin and His Circles by Richard S Varga, Amer. Math. Monthly 113 (4) (2006), 379-381.
3. J Fischer and S Fujino, On Some Contributions of S A Gerschgorin, Information 2 (1999), 257-266.
4. S Fujino and J Fischer, Über S A Gerschgorin (1901-1933), GAMM-Mitt. Ges. Angew. Math. Mech. 21 (1) (1998), 15-19.
5. Obituary: Semyon Aronovich Gershgorin (Russian), Applied Mathematics and Mechanics 1 (1) (1933), 4.
6. G J Tee, Semyon Aronovich Gershgorin, IMAGE 32 (2004), 2-5.
7. G J Tee, Review: Gersgorin and His Circles by Richard S Varga, New Zealand Mathematical Society Newsletter 98 (2006), 21-22.