In the theory of quadratic forms, a Hermite reduction of a real positive definite form is another real positive definite form integrally equivalent to it whose coefficients are reasonably small in the sense defined below.

A positive definite form

${\displaystyle Q(x)=\sum _{i=1}^{n}\sum _{j=1}^{n}Q_{ij}x_{i}x_{j}}$

on ${\displaystyle \mathbb {R} ^{n}}$ is Hermite reduced if the following recursively defined condition is satisfied.

• ${\displaystyle 0<|Q_{11}|\leq (4/3)^{(n-1)/2}{\sqrt[{n}]{\det Q}}}$
• ${\displaystyle 2|Q_{1i}|\leq |Q_{11}|\qquad (i=2,\dots ,n)}$
• The form ${\displaystyle Q'(x_{2},\dots ,x_{n})=Q_{11}Q(x)-(Q_{11}x_{1}+\cdots +Q_{1n}x_{n})^{2}}$ is a Hermite reduced form on ${\displaystyle \mathbb {R} ^{n-1}}$

For every positive definite form ${\displaystyle Q}$ on ${\displaystyle \mathbb {R} ^{n}}$, there exists a ${\displaystyle \mathbb {Z} }$-module isomorphism ${\displaystyle U\colon \mathbb {Z} ^{n}\to \mathbb {Z} ^{n}}$ and a Hermite reduced form ${\displaystyle {\tilde {Q}}}$ on ${\displaystyle \mathbb {R} ^{n}}$ such that^{[1]: 259 [2]: 210 [3]}

${\displaystyle Q\circ (U\otimes _{\mathbb {Z} }\mathbb {R} )={\tilde {Q}}.}$

In matrix notation, for every real ${\displaystyle n\times n}$ positive definite matrix ${\displaystyle Q}$, there exists an integer ${\displaystyle n\times n}$ invertible matrix ${\displaystyle U}$ (so-called unimodular matrix) and an ${\displaystyle n\times n}$ Hermite reduced matrix ${\displaystyle {\tilde {Q}}}$ such that

${\displaystyle U^{T}QU={\tilde {Q}}.}$

Then ${\displaystyle {\tilde {Q}}}$ is called a Hermite reduction of ${\displaystyle Q}$.

Each real positive definite form has only a finite number of Hermite reductions; they are not unique in general.

The Hermite reduction of a binary or ternary positive definite form with integer coefficients with determinant 1 is simply the sum of squares. This is used in a proof of Legendre's three-square theorem: to show that an integer is a sum of squares of three integers it is sufficient to show that it can be represented by a ternary positive definite form with determinant 1.

The Hermite reduction is named after Charles Hermite.

• "Quadratic forms, reduction of", Encyclopedia of Mathematics, EMS Press, 2001 [1994]