**Jacques Salomon Hadamard** FRS (December 8, 1865 – October 17, 1963) was a

FrenchThe French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

who made major contributions in

number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, complex function theory, differential geometry and partial differential equations.

## Biography

The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard attended the

Lycée CharlemagneThe Lycée Charlemagne is located in the Marais quarter of the 4th arrondissement of Paris, the capital city of France.Constructed many centuries before it became a lycée, the building originally served as the home of the Order of the Jesuits...

and

Lycée Louis-le-GrandThe Lycée Louis-le-Grand is a public secondary school located in Paris, widely regarded as one of the most rigorous in France. Formerly known as the Collège de Clermont, it was named in king Louis XIV of France's honor after he visited the school and offered his patronage.It offers both a...

, where his father taught. In 1884 Hadamard entered the

École Normale SupérieureThe École normale supérieure is one of the most prestigious French grandes écoles...

, having been placed first in the entrance examinations both there and at the

École PolytechniqueThe École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...

. His teachers included

TanneryPaul Tannery was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose Notions Mathématiques he contributed an historical chapter...

, Hermite, Darboux,

AppellPaul Appell , also known as Paul Émile Appel, was a French mathematician and Rector of the University of Paris...

, Goursat and

PicardCharles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...

. He obtained his doctorate in 1892 and in the same year was awarded the

*Grand Prix des Sciences Mathématiques* for his essay on the

Riemann zeta function.

In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the

University of BordeauxUniversity of Bordeaux is an association of higher education institutions in and around Bordeaux, France. Its current incarnation was established 21 March 2007. The group is the largest system of higher education schools in southwestern France. It is part of the Academy of Bordeaux.There are seven...

, where he proved his

celebrated inequalityIn mathematics, Hadamard's inequality, first published by Jacques Hadamard in 1893, is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors...

on

determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

s, which led to the discovery of

Hadamard matricesIn mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal...

when equality holds. In 1896 he made two important contributions: he proved the

prime number theoremIn number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

, using complex function theory (also proved independently by de la Vallée Poussin); and he was awarded the Bordin Prize of the

French Academy of SciencesThe French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

for his work on geodesics in the

differential geometry of surfacesIn mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....

and dynamical systems. In the same year he was appointed Professor of Astronomy and Rational Mechanics in Bordeaux. His foundational work on geometry and

symbolic dynamicsIn mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator...

continued in 1898 with the study of geodesics on surfaces of negative curvature. For his cumulative work, he was awarded the Prix Poncelet in 1898.

After the

Dreyfus affairThe Dreyfus affair was a political scandal that divided France in the 1890s and the early 1900s. It involved the conviction for treason in November 1894 of Captain Alfred Dreyfus, a young French artillery officer of Alsatian Jewish descent...

, which involved him personally because his wife was related to Dreyfus, Hadamard became politically active and a staunch supporter of Jewish causes though he professed to be an atheist in his religion.

In 1897 he moved back to Paris, holding positions in the

SorbonneThe Sorbonne is an edifice of the Latin Quarter, in Paris, France, which has been the historical house of the former University of Paris...

and the

Collège de FranceThe Collège de France is a higher education and research establishment located in Paris, France, in the 5th arrondissement, or Latin Quarter, across the street from the historical campus of La Sorbonne at the intersection of Rue Saint-Jacques and Rue des Écoles...

, where he was appointed Professor of Mechanics in 1909. In addition to this post, he was appointed to chairs of analysis at the

École PolytechniqueThe École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...

in 1912 and at the École Centrale in 1920, succeeding

JordanMarie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...

and Appell. In Paris Hadamard concentrated his interests on the problems of mathematical physics, in particular partial differential equations, the

calculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

and the foundations of

functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

. He introduced the idea of

*well-posed problem*The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that# A solution exists# The solution is unique...

and the

*method of descent* in the theory of

partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s, culminating in his seminal book on the subject, based on lectures given at

Yale UniversityYale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...

in 1922. He was elected to the

French Academy of SciencesThe French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

in 1916, in succession to

PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

, whose complete works he helped edit. Later in his life he wrote on

probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and mathematical education. He was awarded the

CNRS Gold medalThe highest scientific research award in France, which is presented annually by the French National Centre for Scientific Research , is the CNRS Gold medal, first awarded in 1954. Past recipients of the Gold medal are* 2011 Jules A...

for his lifetime achievements in 1956.

Hadamard's students included

Maurice FréchetMaurice Fréchet was a French mathematician. He made major contributions to the topology of point sets and introduced the entire concept of metric spaces. He also made several important contributions to the field of statistics and probability, as well as calculus...

,

Paul LévyPaul Pierre Lévy was a Jewish French mathematician who was active especially in probability theory, introducing martingales and Lévy flights...

,

Szolem Mandelbrojt and

André WeilAndré Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...

.

## On creativity

In his book

*Psychology of Invention in the Mathematical Field*, Hadamard uses

introspectionIntrospection is the self-observation and reporting of conscious inner thoughts, desires and sensations. It is a conscious and purposive process relying on thinking, reasoning, and examining one's own thoughts, feelings, and, in more spiritual cases, one's soul...

to describe mathematical thought processes. In sharp contrast to authors who identify

languageLanguage may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...

and

cognitionIn science, cognition refers to mental processes. These processes include attention, remembering, producing and understanding language, solving problems, and making decisions. Cognition is studied in various disciplines such as psychology, philosophy, linguistics, and computer science...

, he describes his own mathematical thinking as largely wordless, often accompanied by mental images that represent the entire solution to a problem. He surveyed 100 of the leading physicists of the day (approximately 1900), asking them how they did their work.

Hadamard described the experiences of the mathematicians/theoretical physicists

Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

,

Hermann von HelmholtzHermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...

,

Henri PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

and others as viewing entire solutions with “sudden spontaneousness.”

Hadamard described the process as having four steps of the five-step

Graham WallasGraham Wallas was an English socialist, social psychologist, educationalist, a leader of the Fabian Society and a co-founder of the London School of Economics....

creativeCreativity refers to the phenomenon whereby a person creates something new that has some kind of value. What counts as "new" may be in reference to the individual creator, or to the society or domain within which the novelty occurs...

process model, with the first three also having been put forth by Helmholtz: Preparation, Incubation, Illumination, and Verification.

## See also

- Cartan–Hadamard theorem
The Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point...

- Cauchy–Hadamard theorem
- Hadamard product:
- entry-wise matrix multiplication
- an infinite product expansion for the Riemann zeta function

- Hadamard code
The Hadamard code is an error-correcting code that is used for error detection and correction when transmitting messages over very noisy or unreliable channels....

- Hadamard's dynamical system
In physics and mathematics, the Hadamard dynamical system or Hadamard's billiards is a chaotic dynamical system, a type of dynamical billiards...

- Hadamard's inequality
In mathematics, Hadamard's inequality, first published by Jacques Hadamard in 1893, is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors...

- Hadamard's method of descent
In mathematics, the method of descent is the term coined by the French mathematician Jacques Hadamard as a method for solving a partial differential equation in several real or complex variables, by regarding it as the specialisation of an equation in more variables, constant in the extra parameters...

- Hadamard finite part integral
In mathematics, Hadamard regularization is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by...

- Hadamard manifold
In mathematics, a Hadamard manifold, named after Jacques Hadamard — sometimes called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold that is complete and simply-connected, and has everywhere non-positive sectional curvature.-Examples:* The real line R with its...

- Hadamard matrix
In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal...

- Hadamard's maximal determinant problem
- Hadamard space
- Hadamard three-circle theorem
In complex analysis, a branch of mathematics, theHadamard three-circle theorem is a result about the behavior of holomorphic functions.Let f be a holomorphic function on the annulusr_1\leq\left| z\right| \leq r_3....

- Hadamard Transform
The Hadamard transform is an example of a generalized class of Fourier transforms...

- Hadamard–Rybczynski equation
- Ostrowski–Hadamard gap theorem
In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic...

## Further reading

.