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220 ALGEBRA

5.7.4-2. Extremal properties of quadratic forms.

A point x 0 on a smooth surface S is called a stationary point of a differentiable function f
defined on S if the derivative of f at the point x 0 in any direction on S is equal to zero. The
value f(x 0 ) of the function f at a stationary point x 0 is called its stationary value.
The unit sphere in a Euclidean space V is the set of all x V such that

x ⋅ x = 1 ( x = 1). (5.7.4.1)

THEOREM. Let B(x, x) be a real quadratic form and let B(x, y)=(Ax) ⋅ y be the
corresponding polar bilinear form, where A is a Hermitian operator. The stationary values
of the quadratic form B(x, x) on the unit sphere (5.7.4.1) coincide with eigenvalues of the
operator A. These stationary values are attained, in particular, on the unit eigenvectors e k
of the operator A.
Remark. If the eigenvalues of the operator A satisfy the inequalities λ 1 ≥ ... ≥ λ n,then λ 1 and λ n are
the largest and the smallest values of B(x, x) on the sphere x ⋅ x = 1.

5.7.5. Second-Order Hypersurfaces
5.7.5-1. Definition of a second-order hypersurface.

A second-order hypersurface in an n-dimensional Euclidean space V is the set of all points
x V satisfying an equation of the form

A(x, x)+ 2B(x)+ c = 0, (5.7.5.1)

where A(x, x) is a real quadratic form different from identical zero, B(x) is a linear form,
and c is a real constant. Equation (5.7.5.1) is called the general equation of a second-order
hypersurface.
Suppose that in some orthonormal basis i 1 , ... , i n ,we have

n n
T
A(x, x)= X AX = a ij x i x k , B(x)= BX = b i x i ,
i,j=1 i=1
T
X =(x 1 , ... , x n ), A ≡ [a ij ], B =(b 1 , ... , b n ).
Then the general equation (5.7.5.1) of a second-order hypersurface in the Euclidean space V
with the given orthonormal basis i 1 , ... , i n can be written as

T
X AX + 2BX + c = 0.
T
The term A(x, x)= X AX is called the group of the leading terms of equation (5.7.5.1),
and the terms B(x)+ c = BX + c are called the linear part of the equation.

5.7.5-2. Parallel translation.
A parallel translation in a Euclidean space V is a transformation defined by the formulas

◦

X = X + X, (5.7.5.2)
◦
where X is a fixed point, called the new origin.