Quadratic Form
quadratic form
[kwä′drad·ik ′fȯrm]Quadratic Form
a second-degree form in n variables x1, x2. …, xn 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 aij = aji. 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, …, xn by other variables y1, y2 … yn, 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 x̅j is the complex conjugate of xj. If such a quadratic form assumes only real values (this is true when aij = āji), 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.