X12+X22=0
is degenerate since it reduces to a single point: the origin. However the correspond- ing matrixAis the identity matrix. On the other hand, over the field of complex numbers, the same equation is that of a non-degenerate quadric, since it factors as
(X1+iX2)(X1−iX2)=0
that is, the equation of two intersecting lines.
Our definition of a quadric in terms of an equation in a given affine basis suggests at once the question:
Can a quadric have several equations in the same affine basis?
The answer is of course yes: it suffices to multiply an equation by a non-zero con- stant! So a more sensible question should rather be:
Are two equations of a quadric in the same affine basis necessarily proportional?
The answer is of course no: both equations
X12+X22=0, X21+2X22=0
describe the same quadric ofR2, namely, the origin. However, the answer is yes for non-degenerate quadrics. We shall prove this result later (see Theorem2.26.9).
2.24 The Reduced Equation of a Quadric
We recall the following
Convention In this section,Kis a field of characteristic distinct from 2 and equa- tion of a quadric always means equation of degree 2.
We are going to show that an appropriate choice of affine basis allows us to considerably simplify the equation of an arbitrary quadric.
Lemma 2.24.1 LetQ⊆Ebe a quadric in a finite dimensional affine space(E, V ).
There exists an affine basis(O;e1, . . . , en)in which the quadric has an equation of the form
n i=1
aiX2i + n i=1
biXi+c=0.
Proof Fix an arbitrary affine basis (O;e1, . . . , en) in which the quadric has the equation (see Proposition2.23.3)
−
→X
t
A−→ X+−→
bt−→
X+c=0
withAa symmetric matrix. The following mapping is thus a symmetric bilinear form onV (see DefinitionG.1.1)
ϕ:V ×V −→K, (v, w)→−→ X
t
A−→ Y
whereX,Yindicate the coordinates ofv,wwith respect to the basise1, . . . , en. Applying Corollary G.2.8, we have another basis e1, . . . , en in which the matrix ofϕ is diagonal. WritingM for the matrix of change of coordinates, the equation of the quadric with respect to the affine basis(O;e1, . . . , en) thus becomes (see CorollaryG.1.4)
−
→XtMtAM−→ X +−→
btM−→
X +c=0
where the matrixA=MtAMis now diagonal.
Theorem 2.24.2 Let Q⊆E be a quadric in a finite dimensional affine space (E, V ). There exists an affine basis(0;e1, . . . , en)with respect to which the equa- tion of the quadric takes one of the following forms:
Type 1 n
i=1aiX2i =1;
Type 2 n
i=1aiX2i =0;
Type 3 n−1
i=1aiX2i =Xn.
Such an equation is called a reduced equation of the quadric.
Proof Let us begin with an affine basis(P;ε1, . . . , εn)with respect to which the equation of the quadric has the form
n i=1
αiYi2+ n i=1
βiYi+γ=0
(see Lemma2.24.1), whereYi indicates the coordinates of a point.
If all coefficientsαi=0 are zero, the equation reduces to an equation of lower degree, namely
n i=1
βiYi+γ=0.
If further, all coefficientsβi are zero, we end up with the equationγ =0: this is an equation of type 2 whenγ =0 (the equation of the whole space) and—up to division byγ—the equation 1=0 of type 1 whenγ=0 (the equation of the empty set). If not all coefficientsβiare zero, we get the equation of a hyperplane; choosing
2.24 The Reduced Equation of a Quadric 95 a new basis with its origin and then−1 first vectors in this hyperplane, the equa- tion becomesZn=0 (the last coordinate): this is an equation of type 3. Notice in particular that when allαi are zero, the quadric is degenerate.
If not all coefficientsαi are zero, up to a possible renumbering of the vectors of the basis, there is no loss of generality in assuming that
αi=0 fori≤m, αi=0 fori > m, 1≤m≤n.
Notice that the casem=nis the case where allαi’s are non-zero.
Let us study the effect of a change of origin, keeping the same basisε1, . . . , εn
ofV. If the new originOhas coordinatesδi and the new coordinates are writtenZi, the change of basis formulổ are simply
Yi=Zi+δi and the equation of the quadric becomes
m i=1
αiZi2+ m i=1
(2αiδi+βi)Zi+ n i=m+1
βiZi+ m
i=1
αiδ2i + n i=1
βiδi+γ=0.
For every index 1≤i≤m, the coefficient ofZi is an expression of degree 1 inδi, withαi=0 as coefficient ofδi. Choosing
δi= − βi 2αi
, i=1, . . . , m yields the equation
m i=1
αiZ2i + n i=m+1
βiZi− m i=1
βi2 4αi +
n i=m+1
βiδi+γ=0. (∗) This time, each variableZi appears at most once: in degree 2 or in degree 1.
Write
k= − m i=1
βi2 4αi +
n i=m+1
βiδi+γ for the constant in this equation.
Whenm=n, equation (∗) becomes n i=1
αiZi2+k=0.
Ifk=0, this is an equation of type 2. Ifk=0, dividing by−kyields an equation of type 1. In those cases, the theorem is proved.
Let us now consider the remaining case:m < n. If βi =0 for alli≥m, then again equation (∗) takes the form
n i=1
αiZi2+k=0
and the same conclusions apply, whatever the values given toδm+1, . . . , δn. Finally, we address the remaining case wherem < n, withβi=0 for somei≥m.
Again, up to a possible renumbering of the last vectors of the basis, there is no loss of generality in assuming thatβn=0. Fix then arbitrary values (for example, 0) for δi, for allm+1≤i≤n−1; it remains to chooseδn. Butkis now an expression of degree 1 inδn, withβn=0 as coefficient ofδn. Choose forδnthe “root” ofk=0, regarded as an equation of degree 1 inδn. Equation (∗) becomes
m i=1
αiZi2+ n i=m+1
βiZi=0.
To conclude the proof, it remains to find another affine basis—with respect to which the coordinates will be writtenXi—and giving rise to a change of coordinates with the properties
Xi=Zi for 1≤i≤m, Xn= − n
i=m+1
βiZi
. We thus want a matrix of change of coordinates which has the form
⎛
⎜⎝ X1
... Xn
⎞
⎟⎠=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎝
1 0
. .. 0
0 1
0ã ã ã0−βm+1ã ã ã −βn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎠
⎛
⎜⎝ Z1
... Zn
⎞
⎟⎠.
But a matrixM is the matrix of a change of coordinates if and only if it is in- vertible. Themfirst lines of the matrix above are trivially linearly independent; and sinceβn=0, so too is the system of themfirst lines together with the last one.
To obtain the expected matrix of change of coordinates, it remains to complete this sequence ofm+1 linearly independent lines to a system ofnlinearly independent lines: this is simply completing a basis to a system ofm+1 linearly independent
vectors ofKn.