quadratic form

quadratic form (plural quadratic forms)

1. (mathematics, number theory, algebra) A homogeneous polynomial of degree 2 in a given number of variables.
   • 1980, John Gregory, Quadratic Form Theory and Differential Equations, Academic Press, page ix:

     Historically, quadratic form theory has been treated as a rich but misunderstood uncle.

   • 2004, Nikita A. Karpenko, Izhboldin's Results on Stably Birational of Equivalence Quadrics, Oleg T. Izhboldin, Geometric Methods in the Algebraic Theory of Quadratic Forms, Springer, Lecture Notes in Mathematics 1835, page 156,

     We claim that this quadratic form is isotropic, and this gives what we need according to [2, Proposition 4.4].

   • 2009, R. Parimala, V. Suresh, J.-P. Tignol, On the Pfister Number of Quadratic Forms, Ricardo Baeza, Quadratic Forms—Algebra, Arithmetic, and Geometry, American Mathematical Society, page 327,

     The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of −1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n − 2 terms and not less. The number of Pfister forms required to express a quadratic form of degree 6 with trivial discriminant is determined in various cases.

   • 2009, W. A. Coppel, Number Theory: An Introduction to Mathematics, 2nd edition, Springer, page 291:

     The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange's theorem that every positive integer is a sum of four squares, was continued in the 19th century by Dirichlet, Hermite, H. J. S. Smith, Minkowski and others.

2. (statistics, multivariate statistics) A scalar quantity of the form ${\displaystyle \varepsilon ^{T}\Lambda \varepsilon }$, where ${\displaystyle \varepsilon }$ is a vector of n random variables, and ${\displaystyle \Lambda }$ is an n-dimensional symmetric matrix.
   • 2009, Allan Gut, An Intermediate Course in Probability, 2nd edition, Springer, page 136:

     Quadratic forms of normal random vectors are of great importance in many branches of statistics, such as least-squares methods, the analysis of variance, regression analysis, and experimental design.

• (mathematics): Quadratic forms in 1, 2 and 3 variables are respectively called unary, binary and ternary quadratic forms.
• (statistics): ${\displaystyle \varepsilon ^{T}\Lambda \varepsilon }$ is said to be a quadratic form in ${\displaystyle \varepsilon }$.