Editing Quadric (section)

==Definition and basic properties==
An ''affine quadric'' is the set of [[zero of a function|zeros]] of a polynomial of degree two. When not specified otherwise, the polynomial is supposed to have [[real number|real]] coefficients, and the zeros are points in a [[Euclidean space]]. However, most properties remain true when the coefficients belong to any [[field (mathematics)|field]] and the points belong in an [[affine space]]. As usual in [[algebraic geometry]], it is often useful to consider points over an [[algebraically closed field]] containing the polynomial coefficients, generally the [[complex number]]s, when the coefficients are real.

Many properties becomes easier to state (and to prove) by extending the quadric to the [[projective space]] by [[projective completion]], consisting of adding [[points at infinity]]. Technically, if
:<math>p(x_1, \ldots,x_n)</math>
is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by [[homogenization of a polynomial|homogenizing]] {{mvar|p}} into
:<math>P(X_0, \ldots, X_n)=X_0^2\,p\left(\frac {X_1}{X_0}, \ldots,\frac {X_n}{X_0}\right)</math>
(this is a polynomial, because the degree of {{mvar|p}} is two). The points of the projective completion are the points of the projective space whose [[projective coordinates]] are zeros of {{mvar|P}}.

So, a ''projective quadric'' is the set of zeros in a projective space of a [[homogeneous polynomial]] of degree two.

As the above process of homogenization can be reverted by setting {{math|1=''X''{{sub|0}} = 1}}:
:<math>p(x_1, \ldots, x_n)=P(1, x_1, \ldots, x_n)\,,</math>
it is often useful to not distinguish an affine quadric from its projective completion, and to talk of the ''affine equation'' or the ''projective equation'' of a quadric.  However, this is not a perfect equivalence; it is generally the case that <math>P(\mathbf{X}) = 0</math> will include points with <math>X_0 = 0</math>, which are not also solutions of <math>p(\mathbf{x}) = 0</math> because these points in projective space correspond to points "at infinity" in affine space.

===Equation===
A quadric in an [[affine space]] of dimension {{mvar|n}} is the set of zeros of a polynomial of degree 2. That is, it is the set of the points whose coordinates satisfy an equation
:<math>p(x_1,\ldots,x_n)=0,</math>
where the polynomial {{mvar|p}} has the form
:<math>p(x_1,\ldots,x_n) = \sum_{i=1}^n \sum_{j=1}^n a_{i,j}x_i x_j + \sum_{i=1}^n (a_{i,0}+a_{0,i})x_i +  a_{0,0}\,,</math>
for a matrix <math>A = (a_{i,j})</math> with <math>i</math> and  <math>j</math> running from 0 to <math>n</math>.  When the [[characteristic (algebra)|characteristic]] of the [[field (mathematics)|field]] of the coefficients is not two, generally <math>a_{i,j} = a_{j,i}</math> is assumed; equivalently <math>A = A^{\mathsf T}</math>.  When the characteristic of the field of the coefficients is two, generally <math>a_{i,j} = 0</math> is assumed when <math>j < i</math>; equivalently <math>A</math> is [[upper triangular]].

The equation may be shortened, as the matrix equation
:<math>\mathbf x^{\mathsf T}A\mathbf x=0\,,</math>
with
:<math>\mathbf x = \begin {pmatrix}1&x_1&\cdots&x_n\end{pmatrix}^{\mathsf T}\,.</math>
The equation of the projective completion is almost identical:
:<math>\mathbf X^{\mathsf T}A\mathbf X=0,</math>
with
:<math>\mathbf X = \begin {pmatrix}X_0&X_1&\cdots&X_n\end{pmatrix}^{\mathsf T}.</math>

These equations define a quadric as an [[hypersurface|algebraic hypersurface]] of [[dimension]] {{math|''n'' − 1}}  and degree  two in a space of dimension {{mvar|n}}.

A quadric is said to be '''non-degenerate''' if the matrix <math>A</math> is [[invertible matrix|invertible]].

A non-degenerate quadric is non-singular in the sense that its projective completion has no [[singular point of an algebraic variety|singular point]] (a cylinder is non-singular in the affine space, but it is a degenerate quadric that has a singular point at infinity).

The singular points of a degenerate quadric are the points whose projective coordinates belong to the [[null space]] of the matrix {{mvar|A}}.

A quadric is reducible if and only if the [[rank (linear algebra)|rank]] of {{mvar|A}}  is one (case of a double hyperplane) or two (case of two hyperplanes).