Questions tagged [quadratic-forms]

A function $f : \mathbb{R^n}\to \mathbb{R^n}$ of the form $$f(x)=x^TAx=\sum_{i,j=1}^na_{ij}x_ix_j\qquad \text{where $A=(a_{ij})_{m \times n}$ be a real symmetric matrix and $x=x(x_1,x_2,...,x_n)\in \mathbb{R}$}$$ is called a quadratic form.

It is usually denoted by $Q(x)$.

Classification of the quadratic form $Q(x)=x^TAx$:

A quadratic form is said to be:

$a:\quad$ negative definite$: Q < 0$ when $x\neq 0$

$b:\quad$ negative semi definite$: Q ≤ 0$ for all $x$ and $Q = 0$ for some $x \neq 0$

$c:\quad$ positive definite$: Q > 0$ when $x \neq 0$

$d:\quad$ positive semi definite$: Q ≥ 0$ for all $x$ and $Q = 0$ for some $x \neq 0$

$e:\quad$ indefinite$: Q > 0$ for some $x$ and $Q < 0$ for some other $x$

Applications: The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers $\mathbb{Z}$ or the $p-$adic integers $\mathbb{Z}_p$. Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in $n$ variables has important applications to algebraic topology.

References:

"https://en.wikipedia.org/wiki/Quadratic_form"

"http://mathworld.wolfram.com/QuadraticForm.html"