What Did Charles Hermite Invent?

Charles Hermite was a renowned French mathematician who made significant contributions to the field of mathematics in the 19th century. He is particularly known for his work in algebra, quadratic forms, and the development of Hermite polynomials and Hermite normal form.

Hermite's Work in Algebra

Hermite's primary contributions to algebra include his work on the theory of quadratic forms. He studied the properties of quadratic forms, which are polynomial expressions of the form ax^2 + bxy + cy^2, where a, b, and c are coefficients. Hermite's research in this area led to the development of the Hermite normal form, a unique representation of a matrix that is widely used in computational linear algebra.

Hermite Polynomials

One of Hermite's most notable inventions is the Hermite polynomials, a sequence of orthogonal polynomials that are named after him. These polynomials have numerous applications in mathematics, physics, and engineering, particularly in the study of quantum mechanics and the analysis of stochastic processes.

The Hermite polynomials are defined by the recurrence relation:

H_n(x) = x H_{n-1}(x) - (n-1) H_{n-2}(x)

where H_0(x) = 1 and H_1(x) = x. The first few Hermite polynomials are:

• H_0(x) = 1
• H_1(x) = x
• H_2(x) = x^2 - 1
• H_3(x) = x^3 - 3x
• H_4(x) = x^4 - 6x^2 + 3

These polynomials have various properties and applications, including in the study of probability distributions, signal processing, and the numerical solution of partial differential equations.

Hermite's Theorem

Hermite also made important contributions to the theory of quadratic forms, including the development of Hermite's theorem. This theorem states that every positive definite quadratic form in n variables can be reduced to a sum of n squares by a suitable linear transformation.

Hermite's theorem has important applications in number theory, cryptography, and the study of quadratic forms in general.

Conclusion

Charles Hermite's contributions to mathematics have had a lasting impact on the field. His work in algebra, quadratic forms, and the development of Hermite polynomials and Hermite normal form have all become fundamental concepts in modern mathematics. Hermite's legacy continues to inspire and influence mathematicians and scientists around the world.