the distance between a point with coordinates and the origin: A closely related notion with geometric overtones is a quadratic space, which is a pair , with V a vector space over a field K, and a quadratic form on V. See below for the definition of a quadratic form on a vector space. History The study of particular quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form , where x, y are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium B.C. In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta, which includes, among many other things, a study of equations of the form . In particular he considered what is now called Pell's equation, , and found a method for its solution. In Europe this problem was studied by Brouncker, Euler and Lagrange. In 1801 Gauss published Disquisitiones Arithmeticae, a major portion of which was devoted to a complete theory of binary quadratic forms over the integers. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated. Associated symmetric matrix Any matrix determines a quadratic form in variables by where . Example Consider the case of quadratic forms in three variables The matrix has the form The above formula gives So, two different matrices define the same quadratic form if and only if they have the same elements on the diagonal and the same values for the sums and In particular, the quadratic form is defined by a unique symmetric matrix This generalizes to any number of variables as follows. General case Given a quadratic form defined by the matrix the matrix is symmetric, defines the same quadratic form as , and is the unique symmetric matrix that defines So, over the real numbers (and, more generally, over a field of characteristic different from two), there is a one-to-one correspondence between quadratic forms and symmetric matrices that determine them. Real quadratic forms A fundamental question is the classification of the real quadratic form under linear change of variables. Jacobi proved that, for every real quadratic