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224 ALGEBRA

2 .Let p be the number of positive eigenvalues of the matrix A and q the number of
◦
negative ones. Central second-order hypersurfaces admit the following classification. A
hypersurface is called:
a)an (n–1)-dimensional ellipsoid if p = n and sign det 2 A =–1,or q = n and sign det 2 A = 1 ;
∗
det A det A
b)an imaginary ellipsoid if p = n and sign det 2 A = 1,or q = n and sign det 2 A =–1;
det A det A
c)a hyperboloid if 0 < p < n (0 < q < n) and sign det 2 A ≠ 0;
det A
d) degenerate if sign det 2 A = 0.
det A

5.7.5-8. Classification of noncentral second-order hypersurfaces.
1 .Let i 2 , ... , i n be an orthonormal basis in which a noncentral second-order hypersurface
◦
is defined by the equation (called its canonical equation)
p
2
λ i x + 2μx n + c = 0, (5.7.5.6)

i
i=1
where x 1 , ... , x n are the coordinates of x in that basis: p =rank (A).
The equation of any noncentral second-order hypersurface S can be reduced to the
canonical form (5.7.5.6) by the following transformations:
1. If p =rank (A), then after the standard simplification and renumbering the basis vectors,
equation (5.7.5.1) turns into
p p n
λ i x + 2 b x + 2 b x + c = 0.
2

i i
i i
i
i=1 i=1 i=p+1
2. After the parallel translation
⎧
b
x + for k = 1, 2, ... , p,
⎨ k
x = k λ k

k
x k for k = p + 1, p + 2, ... , n,
⎩
the last equation can be represented in the form
p n p
2 b i

λ i x i + 2 b x + c = 0, c = c – .
i i
i=1 i=p+1 i=1 λ i
3. Leaving intact the first p basis vectors and transforming the last basis vectors so that the
n

term b x turns into μx , one reduces the hypersurface equation to the canonical

n
i i
i=p+1
form (5.7.5.6).
2 . Noncentral second-order hypersurfaces admit the following classification.
◦
A hypersurface is called:
a)a paraboloid if μ ≠ 0 and p = n – 1; in this case, the parallel translation in the direction
of the x n -axis by – c yields the canonical equation of a paraboloid
2μ
n–1
2
λ i x + 2μx n = 0;
i
i=1
∗
If a 1 = ... = a n = R, then the hypersurface is a sphere of radius R in n-dimensional space.