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222 ALGEBRA
n

The coefficients b of the linear form B (x )= b x are found from the relations [to

i
i i
i=1
this end, one should use (5.7.5.3)]
n n

b x = b i x i ,
i i
i=1 i=1
and the constant is c = c.

5.7.5-4. Invariants of the general equation of a second-order hypersurface.
An invariant of the general second-order hypersurface equation (5.7.5.1) with respect to
parallel translations and orthogonal transformations of an orthogonal basis is, by definition,
any function f(a 11 , ... , a nn , b 1 , ... , b n , c)ofthe coefficients of this equation that does not
change under such transformations of the space.
THEOREM. The coefficients of the characteristic polynomial of the matrix A of the
A B

quadratic form A(x, x) and the determinant det A of the block matrix A = T are
2
2
B c
invariants of the general second-order hypersurface equation (5.7.5.1).
Remark. The quantities det A,Tr(A), rank (A), and rank (A) are invariants of equation (5.7.5.1).
2
5.7.5-5. Center of a second-order hypersurface.

◦
The center of a second-order hypersurface is a point x such that the linear form B (x )

◦
becomes identically equal to zero after the parallel translation that makes x the new origin.
Thus, the coordinates of the center can be found from the system of equations of the center
of a second-order hypersurface
n
a ij j + b i = 0 (i = 1, 2, ... , n).
◦
x
j=1
If the center equations for a hypersurface S have a unique solution, then S is called a
central hypersurface. If a hypersurface S has a center, then S consists of pairs of points,
each pair being symmetric with respect to the center.

Remark 1. For a second-order hypersurface S with a center, the invariants det A,det A, and the free
2

term c are related by

det A = c det A.
2
Remark 2. If the origin is shifted to the center of a central hypersurface S, then the equation of that
hypersurface in new coordinates has the form
det A
2
A(x, x)+ = 0.
det A