Let \(k\) be a field with \(\characteristic k\neq 2\) and \(n\geq 1\). A quadratic form in \(n\) variables over \(k\) is a homogeneous polynomial of degree \(2\) in \(x_1,\ldots,x_n\); it can be written uniquely as \[ q(x_1, \ldots, x_n) = \sum_{i} a_{ii}x_i^2 + \sum_{i < j} 2a_{ij}x_ix_j = X^t A X, \qquad X = (x_1,\ldots,x_n)^t, \] where \(A = (a_{ij})\in\Mat_{n\times n}(k)\) is symmetric, \(a_{ji}=a_{ij}\). The matrix \(A\) is the matrix of the form.
The coefficient of \(x_ix_j\) for \(i<j\) is written \(2a_{ij}\) so that \(X^tAX\) holds with \(A\) symmetric: the off-diagonal entries \(a_{ij}\) and \(a_{ji}\) each contribute \(a_{ij}x_ix_j\).
Let \(k=\RR\) and let \(q(X)=X^tAX\) with \(A\) real symmetric. There is an orthogonal matrix \(P\) such that the change of variables \(X = PX'\) gives \[ q = \sum_{i=1}^n \lambda_i {x_i'}^2, \] where \(\lambda_1,\ldots,\lambda_n\) are the eigenvalues of \(A\).
By the spectral theorem there is an orthogonal \(P\) with \(P^tAP=P^{-1}AP=\diag(\lambda_1,\ldots,\lambda_n)\), and \(q(PX')=X'^t(P^tAP)X'\).
Let \(q\) be a real quadratic form with symmetric matrix \(A\), nondegenerate in the sense that \(\det A\neq 0\). The signature of \(q\) is the pair \((p,m)\), where \(p\) and \(m\) are the numbers of positive and negative coefficients \(\lambda_i\) in a diagonalization \(q=\sum_i\lambda_i{x_i'}^2\) obtained by an invertible change of variables.
(Sylvester's law of inertia)
For a nondegenerate real quadratic form, the numbers \(p\) and \(m\) do not depend on the invertible change of variables that diagonalizes it.
For a nondegenerate real quadratic form \(q\) and \(c>0\), the signature determines the level set \(\theset{q = c}\) up to an invertible linear change of variables. In two variables, signature \((2,0)\) gives an ellipse, \((1,1)\) a hyperbola, and \((0,2)\) the empty set. In three variables, signature \((3,0)\) gives an ellipsoid, \((2,1)\) a hyperboloid of one sheet, \((1,2)\) a hyperboloid of two sheets, and \((0,3)\) the empty set.
See [@Art11]. A basis-free quadratic form is a function \(q\) with \(q(\lambda x) = \lambda^2 q(x)\) whose polarization is bilinear.