# Jacques Hadamard

> French mathematician (1865–1963)

**Wikidata**: [Q164425](https://www.wikidata.org/wiki/Q164425)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Jacques_Hadamard)  
**Source**: https://4ort.xyz/entity/jacques-hadamard

## Summary
Jacques Hadamard was a French mathematician renowned for his contributions to number theory, functional analysis, and partial differential equations. Born in 1865, he made lasting impacts through concepts like the Hadamard matrix and inequality, shaping modern mathematics and physics. His work remains foundational in fields such as quantum computing and coding theory.

## Biography
- **Born**: December 8, 1865
- **Nationality**: French
- **Education**: Studied at Lycée Louis-le-Grand and Lycée Charlemagne; graduated from École Normale Supérieure (1884)
- **Known for**: Hadamard matrix, Hadamard inequality, and contributions to number theory
- **Employer(s)**: University of Bordeaux, Collège de France, Columbia University, École Polytechnique
- **Field(s)**: Mathematics, particularly number theory, functional analysis, and partial differential equations

## Contributions
- **Hadamard Matrix (1893)**: Introduced orthogonal matrices with entries of ±1, critical in signal processing and coding theory.
- **Hadamard Inequality (1893)**: Established bounds for determinants, influencing linear algebra and geometry.
- **Prime Number Theorem (1896)**: Independently proved the distribution of prime numbers, a cornerstone of number theory.
- **Hadamard Gate (Quantum Computing)**: Fundamental quantum logic gate derived from his work on matrices.
- **Cauchy-Hadamard Theorem**: Defined the radius of convergence for power series, advancing complex analysis.
- **Hermite-Hadamard Inequality**: Provided integral estimates, widely used in approximation theory.

## FAQs
### What are Jacques Hadamard's most notable mathematical contributions?
Hadamard is best known for the Hadamard matrix, Hadamard inequality, and his proof of the Prime Number Theorem. These works underpin modern coding theory, signal processing, and number theory.

### Where did Hadamard pursue his academic career?
He taught at prestigious institutions including the University of Bordeaux, Collège de France, Columbia University, and École Polytechnique, reflecting his global academic influence.

### What awards recognized Hadamard's achievements?
He received the Grand Cross of the Legion of Honour, the CNRS Gold Medal, and the Poncelet Prize, among others, highlighting his significance in French and international science.

### How did Hadamard influence quantum computing?
His development of the Hadamard gate, a core component in quantum algorithms, bridged 19th-century mathematics with modern quantum theory.

## Why They Matter
Hadamard's work revolutionized mathematics and its applications. His matrices and inequalities are integral to error-correcting codes, GPS technology, and quantum computing. The Prime Number Theorem, proven independently by Hadamard and de la Vallée Poussin, transformed number theory, enabling advancements in cryptography. Without Hadamard, fields like digital communications and quantum information science would lack foundational tools.

## Notable For
- **Hadamard Matrix**: Essential in signal processing and coding theory.
- **Prime Number Theorem Proof**: A milestone in number theory.
- **Academic Leadership**: Held positions at leading global universities.
- **Awards**: Grand Cross of the Legion of Honour, CNRS Gold Medal.
- **Interdisciplinary Impact**: Influenced physics, engineering, and computer science.

## Body

### Early Life and Education
Born on December 8, 1865, Hadamard studied at Lycée Louis-le-Grand and Lycée Charlemagne. He graduated from École Normale Supérieure in 1884, laying the groundwork for his mathematical career.

### Academic Career
Hadamard taught at the University of Bordeaux (1893–1896) before joining Collège de France. He also lectured at Columbia University and École Polytechnique, fostering international academic collaboration.

### Mathematical Contributions
- **Hadamard Matrix (1893)**: These orthogonal matrices with ±1 entries are crucial in compressed sensing and wireless communications.
- **Hadamard Inequality**: Provides determinant bounds, impacting linear algebra and geometric analysis.
- **Prime Number Theorem (1896)**: Hadamard's proof, alongside de la Vallée Poussin, confirmed the asymptotic distribution of primes, vital for cryptography.
- **Cauchy-Hadamard Theorem**: Determines the radius of convergence for power series, essential in complex analysis.

### Legacy in Science and Technology
- **Quantum Computing**: The Hadamard gate, derived from his matrix, is fundamental in quantum algorithms like Shor's algorithm.
- **Error-Correcting Codes**: Hadamard codes, based on his matrices, enable reliable data transmission.
- **GPS Technology**: His work on differential equations underpins precise positioning systems.

### Awards and Recognition
Hadamard received the Grand Cross of the Legion of Honour, CNRS Gold Medal, and Poncelet Prize. He was elected to the French Academy of Sciences and the Royal Society, reflecting his global stature.

### Institutional Affiliations
- **École Normale Supérieure**: Graduated in 1884.
- **Collège de France**: Taught mathematics, contributing to its academic prestige.
- **Columbia University**: Visiting professor, strengthening Franco-American academic ties.

### Interdisciplinary Influence
Hadamard's research transcended pure mathematics, influencing physics (quantum mechanics), engineering (signal processing), and computer science (cryptography). His work exemplifies the unity of mathematical principles across disciplines.

### Historical Context
A contemporary of Henri Poincaré, Hadamard engaged in the vibrant French mathematical community of the late 19th and early 20th centuries. His longevity (1865–1963) allowed him to witness and contribute to pivotal developments in modern mathematics and science.

## References

1. Great Soviet Encyclopedia (1969–1978)
2. BnF authorities
3. Integrated Authority File
4. [Source](https://www.cnrs.fr/fr/personne/jacques-hadamard)
5. Czech National Authority Database
6. list of professors at Collège de France
7. [Source](https://books.google.fr/books?id=Nj_g6o53UZwC&pg=PA253&lpg=PA253)
8. [Source](http://www.w-volk.de/museum/grave84.htm)
9. [Source](https://www.cnrs.fr/fr/talents/cnrs?medal=38)
10. Léonore database
11. [Source](http://www3.huji.ac.il/htbin/hon_doc/doc_search.pl?search)
12. Complete List of Royal Society Fellows 1660-2007
13. Mathematics Genealogy Project
14. International Standard Name Identifier
15. Virtual International Authority File
16. CiNii Research
17. MacTutor History of Mathematics archive
18. Jacques Salomon Hadamard. Biografisch Portaal
19. [Jacques Salomon Hadamard. Léonore database](http://www.culture.gouv.fr/public/mistral/leonore_fr?ACTION=CHERCHER&FIELD_1=COTE&VALUE_1=19800035/408/54579)
20. SNAC
21. KNAW Past Members
22. Brockhaus Enzyklopädie
23. Proleksis Encyclopedia
24. Croatian Encyclopedia
25. Freebase Data Dumps. 2013
26. [Source](https://rac.es/ficheros/doc/01208.pdf)
27. CONOR.SI
28. Autoritats UB
29. La France savante
30. Treccani's Enciclopedia on line
31. LIBRIS. 2012
32. Catalogo of the National Library of India