The matrix A of the linear operator A in the basis{im k} has canonical Jordan form, and the above theorem is also called the theorem on the reduction of a matrix to canonical Jordan form
Trang 15.6.3-6 Canonical form of linear operators.
An element x is called an associated vector of an operator A corresponding to its eigenvalue λ
if for some m≥ 1, we have
(A – λI) mx≠ 0, (A – λI) m+1x =0
The number m is called the order of the associated vector x.
THEOREM Let A be a linear operator in an n-dimensional unitary space V Then there
is a basis{im k}(k =1,2, , l, m =1, 2, , n k , n1+ n2+· · · + n l = n) in V consisting
of eigenvectors and associated vectors of the operator A such that the action of the operator
Ais determined by the relations
Ai1k = λ ki1k (k =1, 2, , l),
Ai m k = λ kim k + im– k 1 (k =1,2, , l, m =2, 3, , n k)
Remark 1. The vectors i1k (k = 1 , 2, , l) are eigenvectors of the operator A corresponding to the
eigenvalues λ k.
Remark 2. The matrix A of the linear operator A in the basis{im k} has canonical Jordan form, and the above theorem is also called the theorem on the reduction of a matrix to canonical Jordan form.
5.7 Bilinear and Quadratic Forms
5.7.1 Linear and Sesquilinear Forms
5.7.1-1 Linear forms in a unitary space
A linear form or linear functional on V is a linear operator A in L(V, C), where C is the
complex plane
THEOREM For any linear form f in a finite-dimensional unitary space V, there is a
unique element h inV such that
f(x) = x⋅h for all xV.
Remark This statement is true also for a Euclidean spaceV and a real-valued linear functional.
5.7.1-2 Sesquilinear forms in unitary space
A sesquilinear form on a unitary space V is a complex-valued function B(x, y) of two
arguments x, yV such that for any x, y, z in V and any complex scalar λ, the following
relations hold:
1 B(x + y, z) = B(x, z) + B(y, z).
2 B(x, y + z) = B(x, y) + B(x, z).
3 B(λx, y) = λB(x, y).
4 B(x, λy) = ¯ λB(x, y).
Trang 2Remark. Thus, B(x, y) is a scalar function that is linear with respect to its first argument and antilinear
with respect to its second argument For a real spaceV, sesquilinear forms turn into bilinear forms (see
Paragraph 5.7.2).
THEOREM Let B(x, y) be a sesquilinear form in a unitary space V Then there is a
unique linear operator A in L(V, V) such that
B(x, y) = x⋅(Ay).
COROLLARY.If B(x, y) is a sesquilinear form in a unitary space V , then there is a unique linear operator A in L(V, V) such that
B(x, y) = (Ax)⋅y.
5.7.1-3 Matrix of a sesquilinear form
Any sesquilinear form B(x, y) on an n-dimensional linear space with a given basis e1, ,
encan be uniquely represented as
B(x, y) =
n
i,j=1
b ij ξ i ¯η j, b ij = B(e i, ej),
and ξ i , η j are the coordinates of x and y in the given basis The matrix B≡ [b ij] of size
n×n is called the matrix of the sesquilinear form B(x, y) in the given basis e1, , e n This sesquilinear form can also be represented as
B(x, y) = X T BY, X T ≡(ξ1, , ξ n), Y T ≡(¯η1, , ¯η n)
5.7.2 Bilinear Forms
5.7.2-1 Definition of a bilinear form
A bilinear form on a real linear space V is a real-valued function B(x, y) of two arguments
xL, y V satisfying the following conditions for any vectors x, y, and z in V and any
real λ:
1 B(x + y, z) = B(x, z) + B(y, z).
2 B(x, y + z) = B(x, y) + B(x, z).
3 B(λx, y) = B(x, λy) = λB(x, y).
THEOREM Let B(x, y) be a bilinear form in a Euclidean space V Then there is a unique
linear operator A in L(V, V) such that
B(x, y) = (Ax)⋅y.
A bilinear form B(x, y) is said to be symmetric if for any x and y, we have
B (x, y) = B(y, x).
A bilinear form B(x, y) is said to be skew-symmetric if for any x and y, we have
B(x, y) = –B(y, x).
Any bilinear form can be represented as the sum of symmetric and skew-symmetric bilinear forms
THEOREM A bilinear form B(x, y) on a Euclidean space V is symmetric if and only if
the linear operator A in the representation (5.6.6.1) is Hermitian (A = A∗)
Trang 35.7.2-2 Bilinear forms in finite-dimensional spaces.
Any bilinear form B(x, y) on an n-dimensional linear space with a given basis e1, , e n
can be uniquely represented as
B(x, y) =
n
i,j=1
b ij ξ i η j, b ij = B(e i, ej),
and ξ i , η j are the coordinates of the vectors x and y in the given basis The matrix B≡[b ij]
of size n×n is called the matrix of the bilinear form in the given basis e1, , e n The bilinear form can also be represented as
B(x, y) = X T BY, X T ≡(ξ1, , ξ n), Y T ≡(η1, , η n)
Remark. Any square matrix B≡[b ij] can be regarded as a matrix of some bilinear form in a given basis
e1, , en If this matrix is symmetric (skew-symmetric), then the bilinear form is symmetric (skew-symmetric).
The rank of a bilinear form B(x, y) on a finite-dimensional linear space L is defined as the rank of the matrix B of this form in any basis: rank B(x, y) = rank (B).
A bilinear form on a finite dimensional spaceV is said to be nondegenerate (degenerate)
if its rank is equal to (is less than) the dimension of the spaceV, i.e., rank B(x, y) = dim V (rank B(x, y) < dim V).
5.7.2-3 Transformation of the matrix of a bilinear form in another basis
Suppose that the transition from a basis e1, , e nto a basis2e1, ,2enis determined by
the matrix U ≡[u ij ] of size n×n, i.e.
2ei =
n
j=1
u ijej (i =1, 2, , n).
THEOREM The matrices B and 2 B of a bilinear form B(x, y) in the bases e1, , e nand
2e1, ,2en, respectively, are related by
2
B = U T BU.
5.7.2-4 Multilinear forms
A multilinear form on a linear space V is a scalar function B(x1, , x p ) of p arguments
x1, , x p V, which is linear in each argument for fixed values of the other arguments.
A multilinear form B(x, y) is said to be symmetric if for any two arguments x land xl
we have
B(x1, , x k , , x l , , x p ) = B(x1, , x l , , x k , , x p)
A multilinear form B(x, y) is said to be skew-symmetric if for any two arguments x land xl
we have
B(x1, , x k , , x l , , x p ) = –B(x1, , x l , , x k , , x p)
Trang 45.7.3 Quadratic Forms
5.7.3-1 Definition of a quadratic form
A quadratic form on a real linear space is a scalar function B(x, x) obtained from a bilinear form B(x, y) for x = y.
Any symmetric bilinear form B(x, y) is polar with respect to the quadratic form B(x, x).
These forms are related by
B(x, y) = 12[B(x + y, x + y) – B(x, x) – B(y, y)].
5.7.3-2 Quadratic forms in a finite-dimensional linear space
Any quadratic form B(x, x) in an n-dimensional linear space with a given basis e1, , e n
can be uniquely represented in the form
B(x, x) =
n
i,j=1
b ij ξ i ξ j, (5.7.3.1)
where ξ i are the coordinates of the vector x in the given basis, and B≡[b ij] is a symmetric
matrix of size n×n, called the matrix of the bilinear form B(x, x) in the given basis This
quadratic form can also be represented as
B (x, x) = X T BX, X T ≡(ξ1, , ξ n)
Remark. Any quadratic form can be represented in the form (5.7.3.1) with infinitely many matrices B
such that B(x, x) = X T BX In what follows, we consider only one of such matrices, namely, the symmetric matrix A quadratic form is real-valued if its symmetric matrix is real.
A real-valued quadratic form B(x, x) is said to be:
a) positive definite (negative definite) if B(x, x) >0(B(x, x) <0) for any x≠ 0;
b) alternating if there exist x and y such that B(x, x) >0and B(y, y) <0;
c) nonnegative (nonpositive) if B(x, x)≥ 0(B(x, x)≤ 0) for all x.
If B(x, y) is a polar bilinear form with respect to some positive definite quadratic form
B(x, x), then B(x, y) satisfies all axioms of the scalar product in a Euclidean space.
Remark The axioms of the scalar product can be regarded as the conditions that determine a bilinear form that is polar to some positive definite quadratic form.
The rank of a quadratic form on a finite-dimensional linear space V is, by definition,
the rank of the matrix of that form in any basis ofV, rank B(x, x) = rank (B).
A quadratic form on a finite-dimensional linear space V is said to be nondegenerate (degenerate) if its rank is equal to (is less than) the dimension of V, i.e., rank B(x, x) = dim V (rank B(x, x) < dim V).
5.7.3-3 Transformation of a bilinear form in another basis
Suppose that the transition from the basis e1, , e nto the basis2e1, ,2enis given by the
matrix U ≡[u ij ] of size n×n, i.e.
2ei =
n
j=1
u ijej (i =1, 2, , n).
Then the matrices B and 2 B of the quadratic form B(x, x) in the bases e1, , e nand
2e1, ,2en, respectively, are related by
2
B = U T BU.
Trang 55.7.3-4 Canonical representation of a real quadratic form.
Let g1, , g n be a basis in which the real quadratic form B(x, x) in a linear space V admits
the representation
B(x, x) =
n
i=1
λ i η i2 (5.7.3.2)
where η1, , η n are the coordinates of x in that basis This representation is called a
canonical representation of the quadratic form, the real coefficients λ1, , λ nare called
the canonical coefficients, and the basis g1, , g n is called the canonical basis.
The number of nonzero canonical coefficients is equal to the rank of the quadratic form THEOREM Any real quadratic form on an n-dimensional real linear space V admits a
canonical representation (5.7.3.2)
1◦ Lagrange method The basic idea of the method consists of consecutive transformations
of the quadratic form: on every step, one should single out the perfect square of some linear form
Consider a quadratic form
B(x, x) =
n
i,j=1
b ij ξ i ξ j. Case 1 Suppose that for some m (1 ≤m≤n), we have b mm ≠ 0 Then, letting
B(x, x) = 1
b mm
k=1
b mk ξ k
2
+ B2(x, x),
one can easily verify that the quadratic form B2(x, x) does not contain the variable ξ m This method of separating a perfect square in a quadratic form can always be applied if the
matrix [b ij ] (i, j =1,2, , n) contains nonzero diagonal elements.
Case 2 Suppose that b mm =0, b ss= 0, but b ms ≠ 0 In this case, the quadratic form can be represented as
B(x, x) = 1
2b sm
k=1
(b mk + b sk )ξ k
2
2b sm
k=1
(b mk – b sk )ξ k
2
+ B2(x, x),
where B2(x, x) does not contain the variables ξ m , ξ s, and the linear forms in square brackets are linearly independent (and therefore can be taken as new independent variables or coordinates)
By consecutive combination of the above two procedures, the quadratic form B(x, x) can
always be represented in terms of squared linear forms; these forms are linearly independent, since each contains a variable which is absent in the other linear forms
2◦ Jacobi method Suppose that
Δ1≡b11≠ 0, Δ2 ≡b11 b12
b21 b22
≠ 0, ., Δn≡det B ≠ 0,
where B≡[b ij ] is the matrix of the quadratic form B(x, x) in some basis e1, , e n One can obtain a canonical representation of this form using the formulas
λ1 =Δ1, λ i = Δi
Δi–1 (i =2,3, , n).
Trang 6The basis e1, , e nis transformed to the canonical basis g1, , g nby the formulas
gi =
n
j=1
α ijej (i =1, 2, , n),
α ij = (–1)i+jΔi–1 ,j
Δi–1 ,
where Δi–1 ,j is the minor of the submatrix of B ≡ [b ij] formed by the elements on the intersection of its rows with indices1,2, , i –1and columns with indices1,2, , j –1,
j+1, i.
5.7.3-5 Normal representation of a real quadratic form
Let g1, , g nbe a basis of a linear spaceV in which the quadratic form B(x, x) is written as
B(x, x) =
n
i=1
ε i η i2 (5.7.3.3)
where η1, , η n are the coordinates of x in that basis, and ε1, , ε n are coefficients taking the values –1,0, or1 Such a representation of a quadratic form is called its normal
representation.
Any real quadratic form B(x, x) in an n-dimensional real linear space V admits a
normal representation (5.7.3.3) Such a representation can be obtained by the following transformations:
1 One obtains its canonical representation (see Paragraph 5.7.3-4):
B(x, x) =
n
i=1
λ i μ2i
2 By the nondegenerate coordinate transformation
η i=
⎧
⎪
⎪
1
√
λ i μ i for λ i >0,
1
√
–λ i μ i for λ i <0,
μ i for λ i =0, the canonical representation turns into a normal representation
LAW OF INERTIA OF QUADRATIC FORMS The number of terms with positive coefficients and the number of terms with negative coefficients in any normal representation of a real quadratic form does not depend on the method used to obtain such a representation
The index of inertia of a real quadratic form is the integer k equal to the number of
nonzero coefficients in its canonical representation (this number coincides with the rank
of the quadratic form) Its positive index of inertia is the integer p equal to the number
of positive coefficients in the canonical representation of the form, and its negative index
of inertia is the integer q equal to the number of its negative canonical coefficients The
integer s = p – q is called the signature of the quadratic form.
A real quadratic form B(x, x) on an n-dimensional real linear space V is
a) positive definite (resp., negative definite) if p = n (resp., q = n);
b) alternating if p≠ 0, q≠ 0;
c) nonnegative (resp., nonpositive) if q =0, p < n (resp., p =0, q < n).
Trang 75.7.3-6 Criteria of positive and negative definiteness of a quadratic form.
1◦ A real quadratic form B(x, x) is positive definite, negative definite, alternating,
non-negative, nonpositive if the eigenvalues λ i of its matrix B ≡ [b ij] are all positive, are all negative, some are positive and some negative, are all nonnegative, are all nonpositive, respectively
2◦ Sylvester criterion A real quadratic form B(x, x) is positive definite if and only if the
matrix of B(x, x) in some basis e1, , e nsatisfies the conditions
Δ1≡b11 >0, Δ2 ≡b11 b12
b21 b22
>0, , Δn≡det B >0
If the signs of the minor determinants alternate,
Δ1 <0, Δ2>0, Δ3<0, ,
then the quadratic form is negative definite
3◦ A real matrix B is nonnegative and symmetric if and only if there is a real matrix C
such that B = C T C.
5.7.4 Bilinear and Quadratic Forms in Euclidean Space
5.7.4-1 Reduction of a quadratic form to a sum of squares
THEOREM1 Let B(x, y) be a symmetric bilinear form on a n-dimensional Euclidean
spaceV Then there is an orthonormal basis i1, , i ninV and there are real numbers λ k
such that for any xV the real quadratic form B(x, x) can be represented as the sum of
squares of the coordinates ξ kof x in the basis i1, , i n:
B(x, x) =
n
k=1
λ k ξ2
k.
THEOREM2 Let A(x, y) and B(x, y) be symmetric bilinear forms in a n-dimensional
real linear spaceV, and suppose that the quadratic form A(x, x) is positive definite Then
there is a basis i1, , i n ofV such that the quadratic forms A(x, x) and B(x, x) can be
represented in the form
A(x, x) =
n
k=1
λ k ξ2
k, B(x, x) =
n
k=1
ξ2
k,
where ξ kare the coordinates of x in the basis i1, , i n The set of real λ1, , λ ncoincides
with the spectrum of eigenvalues of the matrix B– 1A (the matrices A and B can be taken
in any basis), and this set consists of the roots of the algebraic equation
det(A – λB) =0