Page 256 -

5.7. BILINEAR AND QUADRATIC FORMS 223

5.7.5-6. Simplification of a second-order hypersurface equation.

Let A be the operator whose matrix in an orthonormal basis i 1 , ... , i n coincides with
the matrix of a quadratic form A(x, x). Suppose that the transition from the orthonormal
basis i 1 , ... , i n to the orthonormal basis i , ... , i is defined by an orthogonal matrix P,

1
n
–1
and A = P AP is a diagonal matrix with the eigenvalues of the operator A on the main

diagonal. Then the equation of the hypersurface (5.7.5.1) in the new basis takes the form
n n
λ i x + 2 b x + c = 0, (5.7.5.4)
2

i i
i
i=1 i=1
where the coefficients b are determined by the relations

i
n n

b x = b i x i .
i i
i=1 i=1
The reduction of any equation of a second-order hypersurface S to the form (5.7.5.4) is
called the standard simplification of this equation (by an orthogonal transformation of the
basis).
5.7.5-7. Classification of central second-order hypersurfaces.
1 .Let i 2 , ... , i n be an orthonormal basis in which a second-order central hypersurface is
◦
defined by the equation (called its canonical equation)
n 2
x i det A
2
ε i 2 +sign = 0, (5.7.5.5)
a det A
i=1 i
where x 1 , ... , x n are the coordinates of x in that basis, and the coefficients ε 1 , ... , ε n take
the values –1, 0,or 1. The constants a k > 0 are called the semiaxes of the hypersurface.
The equation of any central hypersurface S can be reduced to the canonical equation
(5.7.5.5) by the following transformations:
1. By the parallel translation that shifts the origin to the center of the hypersurface, its
equation is transformed to (see Paragraph 5.7.5-5):

det A
2
A(x, x)+ = 0.
det A
2. By the standard simplification of the last equation, one obtains an equation of the
hypersurface in the form
n
2 det A
2
λ i x + = 0.
i
i=1 det A
3. Letting
| det A|
⎧
1 = ⎨ |λ k | if det A ≠ 0, (k = 1, 2, ... , n),
2
2
a 2 ⎩ | det A| ε k =sign λ k
k |λ k | if det A = 0,
2
one passes to the canonical equation (5.7.5.5) of the central second-order hypersurface.