A second-order hypersurface in an n-dimensional Euclidean space V is the set of all points xV satisfying an equation of the form where Ax, x is a real quadratic form different from ident
Trang 15.7.4-2 Extremal properties of quadratic forms.
A point x0on 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 x0in any direction on S is equal to zero The
value f (x0) of the function f at a stationary point x0is called its stationary value.
The unit sphere in a Euclidean space V is the set of all xV such that
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 ek
of the operator A.
Remark. If the eigenvalues of the operator A satisfy the inequalities λ1≥ .≥λ n , then λ1and λ nare
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
xV satisfying an equation of the form
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 i1, , i n, we have
A (x, x) = X T AX =
n
i,j=1
a ij x i x k, B (x) = BX =
n
i=1
b i x i
X T = (x
1, , x n), A≡[a ij], B = (b1, , b n)
Then the general equation (5.7.5.1) of a second-order hypersurface in the Euclidean spaceV
with the given orthonormal basis i1, , i ncan be written as
X T AX+2BX + c =0
The term A(x, x) = X T 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
whereX ◦ is a fixed point, called the new origin.
Trang 2In terms of coordinates, (5.7.5.2) takes the form
x k = x k+x ◦ k (k =1, 2, , n),
where X T = (x1, , x n ), X T = (x 1, , x n),X ◦ T = (x ◦
1, , x ◦ n).
Under parallel translations any basis remains unchanged
The transformation of the spaceV defined by (5.7.5.2) reduces the hypersurface equation
(5.7.5.1) to
A(x, x) +2B (x ) + c =0,
where the linear form B (x ) and the constant c are defined by
B (x ) = A(x ,◦ x) + B(x ), c = A( ◦x, x) +◦ 2B(◦ x) + c,
or, in coordinate notation,
B (x) ≡B X =n
i=1
b
i x i c =
n
i=1
(b i + b i)x ◦ i + c, b
i =
n
j=1
a ij x ◦ j + b i
Under parallel translation the group of the leading terms preserves its form
5.7.5-3 Transformation of one orthonormal basis into another
The transition from one orthonormal basis i1, , i nto another orthonormal basis i1, , i n
is defined by an orthogonal matrix P ≡[p ij ] of size n×n, i.e.,
i i =
n
j=1
p iji (i =1, 2, , n)
Under this orthogonal transformation, the coordinates of points are transformed as follows:
X = P X,
or, in coordinate notation,
x
k=
n
i=1
p ki x i (k =1,2, , n), (5.7.5.3)
where X T = (x1, , x n ), X T = (x 1, , x n)
If the transition from the orthonormal basis i1, , i nto the orthonormal basis i1, , i n
is defined by an orthogonal matrix P , then the hypersurface equation (5.7.5.1) in the new
basis takes the form
A (x, x) +2B (x ) + c =0
The matrix A ≡[a ij ] (A (x, x ) = X T A X ) is found from the relation
A = P– 1AP.
Thus, when passing from one orthonormal basis to another orthonormal basis, the matrix
of a quadratic form is transformed similarly to the matrix of some linear operator Note
that the operator A whose matrix in an orthonormal basis coincides with the matrix of the
quadratic form A(x, x) is Hermitian.
Trang 3The coefficients b i of the linear form B (x) =n
i=1 b
i x iare found from the relations [to
this end, one should use (5.7.5.3)]
n
i=1
b
i x i =
n
i=1
b i x i
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 (a11, , a nn , b1, , b n , c) of the 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
quadratic form A(x, x) and the determinant det 2 Aof the block matrix 2A = A B
are invariants of the general second-order hypersurface equation (5.7.5.1)
Remark. The quantities det A, Tr(A), rank (A), and rank (2 A) are invariants of equation (5.7.5.1).
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 makesx 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
j=1
a ij x ◦ j + b i =0 (i =1, 2, , n)
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 2 A, and the free
term c are related by
det 2A = c det A.
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
A(x, x) +det 2A
det A =0.
Trang 45.7.5-6 Simplification of a second-order hypersurface equation.
Let A be the operator whose matrix in an orthonormal basis i1, , i n coincides with
the matrix of a quadratic form A(x, x) Suppose that the transition from the orthonormal
basis i1, , i nto the orthonormal basis i1, , i n is defined by an orthogonal matrix P , and A = P–1AP 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
i=1
λ i x
i2+2n
i=1
b
where the coefficients b iare determined by the relations
n
i=1
b
i x i =
n
i=1
b i x i
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 i2, , i nbe an orthonormal basis in which a second-order central hypersurface is
defined by the equation (called its canonical equation)
n
i=1
ε i x
2
i
a2
i
+ signdet 2A
where x1, , x n are the coordinates of x in that basis, and the coefficients ε1, , ε ntake the values –1,0, or1 The constants ak >0are 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):
A(x, x) + det 2A
det A =0
2 By the standard simplification of the last equation, one obtains an equation of the hypersurface in the form
n
i=1
λ i x2
i + det 2det A A =0
3 Letting
1
a2
k
=
⎧
⎨
⎩
|λ k||det A|
|det 2A| if det 2A≠ 0,
|λ k| if det 2A=0,
ε k = sign λ k (k =1,2, , n), one passes to the canonical equation (5.7.5.5) of the central second-order hypersurface
Trang 52◦ 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 det2A
detA= –1, or q = n and sign det2A
detA=1∗;
b) an imaginary ellipsoid if p = n and signdet2A
detA =1, or q = n and sign detA2
detA = –1;
c) a hyperboloid if0< p < n (0 < q < n) and sign det2A
detA ≠ 0;
d) degenerate if signdet2A
detA =0
5.7.5-8 Classification of noncentral second-order hypersurfaces
1◦ Let i2, , i
nbe an orthonormal basis in which a noncentral second-order hypersurface
is defined by the equation (called its canonical equation)
p
i=1
where x1, , 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
i=1
λ i x
i2+2p
i=1
b
i x i+2 n
i=p+1
b
i x i + c =0
2 After the parallel translation
x
k=
⎧
⎨
⎩x
k+ b
k
λ k for k =1,2, , p,
x
k for k = p +1, p +2, , n, the last equation can be represented in the form
p
i=1
λ i x i2+2 n
i=p+1
b
i x i + c =0, c = c –p
i=1
b i
λ i.
3 Leaving intact the first p basis vectors and transforming the last basis vectors so that the
term n
i=p+1 b
i x i turns into μx n, one reduces the hypersurface equation to the canonical
form (5.7.5.6)
2◦ Noncentral second-order hypersurfaces admit the following classification.
A hypersurface is called:
a) a paraboloid if μ≠ 0and p = n –1; in this case, the parallel translation in the direction
of the x n-axis by –2μc yields the canonical equation of a paraboloid
n–1
i=1
λ i x2
i +2μx n=0;
∗ If a1= = a n = R, then the hypersurface is a sphere of radius R in n-dimensional space.
Trang 6b) a central cylinder if μ =0, p < n; its canonical equation has the form
p
i=1
λ i x2
i + c =0;
c) a paraboloidal cylinder if μ≠ 0, p < n –1; in this case, the parallel translation along the
x n-axis by –2μc yields the canonical equation of a paraboloidal cylinder
p
i=1
λ i x2i +2μx n=0
5.8 Some Facts from Group Theory
5.8.1 Groups and Their Basic Properties
5.8.1-1 Composition laws
Let T be a mapping defined on ordered pairs a, b of elements of a set A and mapping each
pair a, b to an element c of A In this case, one says that a composition law is defined on the set A The element cA is called the composition of the elements a, bAand is denoted
by c = aTb.
A composition law is commonly expressed in one of the two forms:
1 Additive form: c = a + b; the corresponding composition law is called addition and c is called the sum of a and b.
2 Multiplicative form: c = ab; the corresponding composition law is called multiplication and c is called the product of a and b.
A composition law is said to be associative if
a T(bTc) = (aTb)Tc for all a, b, cA
In additive form, this relation reads a + (b + c) = (a + b) + c; and in multiplicative form,
a (bc) = (ab)c.
A composition law is said to be commutative if
a Tb = bTa for all a, bA
In additive form, this relation reads a + b = b + a; and in multiplicative form, ab = ba.
An element e of the set A is said to be neutral with respect to the composition law T if
a Te = a for any aA
In the additive case, a neutral element is called a zero element, and in the multiplicative case, an identity element.
An element b is called an inverse of aA if aTb = e The inverse element is denoted
by b = a– 1.
In the additive case, the inverse element of a is called the negative of a and it is denoted
by –a.
Example 1 Addition and multiplication of real numbers are composition laws on the set of real numbers.
Both these laws are commutative The neutral element for the addition is zero The neutral element for the multiplication is unity.
Trang 75.8.1-2 Notion of a group.
A group is a set G with a composition law T satisfying the conditions:
1 The law T is associative.
2 There is a neutral element eG
3 For any aG , there is an inverse element a–1
A group G is said to be commutative or abelian if its composition law T is commutative.
Example 2 The set Z of all integer numbers is an abelian group with respect to addition The set of all
positive real numbers is an abelian group with respect to multiplication Any linear space is an abelian group with respect to the addition of its elements.
Example 3 Permutation groups Let E be a set consisting of finitely many elements a, b, c, A
permutation of E is a one-to-one mapping of E onto itself A permutation f of the set E can be expressed in
the form
f=
a b c .
f (a) f (b) f (c)
.
On the set P of all permutations of E, the composition law is introduced as follows: if f1 and f2are two
permutations of E, then their composition f2◦ f1is the permutation obtained by consecutive application of f1 and f2 This composition law is associative The set of all permutations of E with this composition law is a
group.
Example 4 The group Z2that consists of two elements 0 and 1 with the multiplication defined by
0 ⋅ 0 = 0, 0 ⋅ 1 = 1, 1 ⋅ 0 = 1, 1 ⋅ 1 = 1 and the neutral element 0is called the group of modulo2residues.
Properties of groups:
1 If aTa–1= e, then a–1Ta = e.
2 eTa = a for any a.
3 If aTx = e and aTy = e, then x = y.
4 The neutral element e is unique.
5.8.1-3 Homomorphisms and isomorphisms
Recall that a mapping f : A → B of a set A into a set B is a correspondence that associates each element of A with an element of B The range of the mapping f is the set of all
b B such that b = f (a) One says that f is a one-to-one mapping if it maps different elements of A into different elements of B, i.e., for any a1, a2 A such that a1 ≠a2, we
have f (a1)≠f (a2)
A mapping f : A → B is called a mapping of the set A onto the set B if each element
of B is an image of some element of A, i.e., for any bB , there is aA such that b = f (a).
A mapping f of A onto B is said to be invertible if there is a mapping g : B → A such that g(f (a)) = a for any aA The mapping g is called the inverse of the mapping f and
is denoted by g = f–1
For definiteness, we use the multiplicative notation for composition laws in what follows, unless indicated otherwise
Let G be a group and let 2 G be a set with a composition law A mapping f : G → 2 Gis
called a homomorphism if
f (ab) = f (a)f (b) for all a , bG;
and the subset of 2G consisting of all elements of the form f (a), a G , is called a homo-morphic image of the group G and is denoted by f (G) Note that here the set 2 Gwith a composition law is not necessarily a group However, the following result holds