Quadratic Form (redirected from Binary quadratic form)

quadratic form [kwä′drad·ik ′fȯrm]

(mathematics)

Any second-degree, homogeneous polynomial.

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Quadratic Form 

a second-degree form in n variables x₁, x₂. …, x_n that is, a polynomial of these variables, each term of which contains either the square of one of the variables or the product of two different variables. The general form of a quadratic form for n = 2 is

ax₁² + bx₂² + cx₁x₂

and for n = 3,

ax₁² + bx₂² + bc₃² + dx₁x₂ + ex₁x₃ + fx₂x₃

where a, b, …, fare any number. An arbitrary quadratic form is written as

where it is assumed that a_ij = a_ji. Quadratic forms in two, three, and four variables are directly connected with the theory of second-order curves (in the plane) and surfaces (in space): in Cartesian coordinates the equation of a centrally symmetric second-order curve or surface has the form A(x) = 1, where A (x) is a quadratic form. In homogeneous coordinates the left-hand side of any equation of a second-order curve or surface is a quadratic form. Upon repacement of the variables x₁, x₂, …, x_n by other variables y₁, y₂ … y_n, which are linear combinations of the old variables, a quadratic form transforms into another quadratic form. By means of an appropriate choice of new variables (nonsingular linear transformation), a quadratic form may be reduced to the form of a sum of squares of the variables, multiplied by certain numbers. Moreover, neither the number of squares (the rank of the quadratic form) nor the difference between the number of positive and the number of negative coefficients of the squares (the signature of the quadratic form) depends on the process of reducing the quadratic form to a sum of squares (the Sylvester law of inertia). The indicated reduction may even be accomplished by special (orthogonal) transformations. Such a transformation corresponds geometrically to a reduction of the second-order curve or surface to the principal axes.

If we admit complex variables, then we obtain quadratic forms of the type

where x̅_j is the complex conjugate of x_j. If such a quadratic form assumes only real values (this is true when a_ij = ā_ji), then it is called Hermitian. The basic facts that are valid for real quadratic forms hold true for Hermitian forms: the possibility of reduction to a sum of squares, invariance of the rank, and the law of inertia.

REFERENCE

Mal’tsev, A. I. Osnovy lineinoi algebry, 3rd ed. Moscow, 1970.