an ellipsoid; if all the eigenvalues are negative, then it is an imaginary ellipsoid (we get the equation of an ellipsoid but with imaginary radii); if some eigenvalues are positive and some are negative, then it is a hyperboloid. If there exist one or more eigenvalues , then the shape depends on the corresponding . If the corresponding , then the solution set is a paraboloid (either elliptic or hyperbolic); if the corresponding , then the dimension degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of . When the solution set is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic. Integral quadratic forms Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology. An integral quadratic form has integer coefficients, such as ; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning if . This is the current use of the term; in the past it was sometimes used differently, as detailed below. Historical use Historically there was some confusion and controversy over whether the notion of integral quadratic form should mean: twos in the quadratic form associated to a symmetric matrix with integer coefficients twos out a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal) This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices). In "twos in", binary quadratic forms are of the form , represented by the symmetric matrix this is the convention Gauss uses in Disquisitiones Arithmeticae. In "twos out", binary quadratic forms are of the form , represented by the symmetric matrix Several points of view mean that twos out has been adopted as the standard convention. Those include: better understanding of the 2-adic theory