BILLINIAR FORMS AND QUADRATIC FORMS
§1. Billinear forms
Let V be a vector space over the field K.
It is called billinear form over the vector space V an application g: VVK, which satisfies the conditions:
1) g(αx + βy,z) = α g(x,z) + β g(y,z)
g(x,αy + βz) = α g(x,y) + β g(x,z)
In other words, a billinear form is an application g : VVK, in both of arguments.
Example 1. Canonical scalar product on the vector space Rn
< , > : Rn Rn Rn, having the analytic expression <x,y> = x1y1 + x2y2 + xnyn in the canonical basis B = , is a billinear form.
The set of billinear form defined over the vector space V forms a vector space over K, with respect to the additive and multiplicative operations of functions.
A billinear form g: VVK is called
a)symmetrical if g(x,y) = g(y,x) , x ,y V
b) skew symmetrical if g(x,y)= - g(y,x). x ,y V.
Let be Vn a n-dimensional vector space , B a basis of the vector space Vn and two arbitrary vectors x = and yj = .
The expression of the billinear form of g, for the vectors x and y, will be given by the relation:
g(x,y)=g(,) = . (1.1)
From (1.1) results that a billinear form g is perfectly determined if its values are known on the vector basis B .
Denoting aij = g(ei,ej), where i,j = 1,2, ,n, the expression (1.1) can be written
called the analytical expression of the billinear form g and the matrix A = (aij) is called the matrix of the billinear form of g with respect to the basis B.
If we denote X = and Y = then (1.2) can be written as a matrix as follows :
g(x,y) = tXAY (1.2)
The correlation between any billinear form and a square matrix A, is an isomorphism of vector spaces. Moreover, a symmetrical (skew symmetrical) billinear form in a given basis of the vector space Vn, can be associated with a symmetrical (skew symmetrical) matrix.
If Ω Mn ( K ) is the passing matrix from the basis B to the basis B , in the vector space Vn and A, A' are the associated matrices to the billinear g form with respect to the considered bases, then
A' = tΩAΩ (1.3)
Proof. The expression of the billinear form g in the basis B' is given by
g(x,y) = tX′A′Y′. Using the coordinates transformation relations:
X= Ω tX, Y= Ω tY ; passing from the basis B to the basis B , we obtain
g(x,y) = t XAY =t( X′)A( Y′) = tX(t A )Y′,
and from it, by identification, we obtain the relation (1.3) .
From (1.3) results that rank A' = rank A, because the matrix Ω is non-degenerated .
The rank of the matrix A defines the rank of the billinear form g. This rank is invariable on base change. In these conditions, the notion of non-degenerated (degenerated) billinear form is justified as being that billinear form g:VVK whos associated matrix A is non-degenerated (degenerated), with respect to a base B from the vector space V.
Let g : VVK be a symmetrical billinear form. The set Ker g = is called the Kernel of the billinear form g . (1.4)
1.5 Proposition. The subset Ker g is a vector subspace of V .
Proof. For x, y Ker g there is g(x,z) = 0, g(y,z) = 0 ,z V and for K we obtain g(αx + βy,z) = α g(x,z) + β g(y,z) = 0 , z V , then αx + βy Ker g , meaning that the subset Ker g is a vector subspace.
If g: VVK is a symmetrical billinear form, then rank g + dim Ker g = n. (1.5)
Proof: Consider B a basis of the vector space Vn and A=(aij) the matrix of the billinear form with respect to this basis.
If x = i ei and y = j ej are any two vectors from Vn, then
g(x,y) = ijxiyj = yj .
The relation xKer g is true if and only if x Vn is one of solutions of the linear and homogeneous equations system:
aij xi = 0 , j = 1,2, ,n , (1.6)
meaning that the kernel Ker g is the same as the set of solution for the system (1.6) which is a vector subspace of dimension equal with the number of secondary unknowns of the system. From rank g = rank A, we obtain (1.5).
The symmetrical billinear form g is no degenerated if and only if Ker g =.
§2. Quadratic forms
A map h: VK, defined on a K vector space V, with the property that a symmetrical billinear form g:VVK exists, such that:
h(x) = g(x,x) , xV .
The symmetrical billinear form g which defines uniquely the quadratic form h is called a polar form or a halved form associated to h .
If the quadratic form h is known then the associated polar form is given by the following expression:
g(x,y) = [ h(x + y) h(x) h(y)] (2.2)
Example. The canonical scalar product defined on the arithmetical space Rn uniquely defines the quadratic form
h(x) =< x,x > = x Rn ,
which is the square of Euclidian norm.
Let us consider a finite dimensional vector space Vn, B one of its bases, and x = i ei an arbitrary vector from Vn.
The analytical expression of the quadratic form h is given by
h(x) = g(x,x) = ijxixj = tXAX, (2.3)
where A = (aij), i,j = 1,2, ,n is the associated matrix of the symmetrical billinear form g.
The matrix and the rank of the symmetrical billinear form g defines the matrix, respectively the rank of the quadratic form h .
The vectors x,yV are called orthogonal to the symmetrical billinear form g (or to the quadratic form h) if g(x,y) = 0 .
If UVn is a vector subspace of Vn, then the set U┴ = is a vector subspace of V, called the orthogonal complement of U . Moreover, if g is non-degenerated on the subspace U of the finite dimensional space Vn then, UU┴ =Vn .
The set UV is called orthogonal with respect to the symmetrical billinear form g if any two vectors of the set U are orthogonal, meaning g(x,y) = 0 x,yU, where xy .
If the subset B =Vn is an orthogonal base of the vector space Vn, with respect to g, then the matrix of the billinear form g is a diagonal matrix. Indeed, aij = g(ei,ej) = 0, i j.
In this case, the analytical expression of the symmetrical billinear form g is:
g(x,y) = aii xi yi
and the analytical expression of the quadratic form h is given by
h(x) = aii xi2
The expressions (1.10) and (1.11) are called canonical forms.
§3. The reduction of the expression of the quadratic forms to
the canonical form
Let K be a vector space, h: VnK a quadratic form on Vn and A the symmetrical matrix which represents the quadratic form h with respect to the basis B Vn. The analytical expression of the quadratic form h in this basis is:
h(x)= or using matrices h(x) = tXAX (3.1)
If we change the base of the vector space Vn, the quadratic form h is caracterised by the matrix A' = t, where is the passing matrix from the base B to the base B'. Naturally, there is the problem of finding a base which has the simplest expression with respect to the quadratic form h. If the body K has the caracteristic value different than 2 then the symmetrical matrix A admits a diagonal form, meaning that h admits a canonical form.
Any expression of a quadratic form over a vector space Vn can be reduced to the canonical form, using a change of base.
Proof. Let h(x) = ijxixj be the analytical expression of a non-zero quadratic form h. First we consider the case aii = 0 , i = . Because h is not identically zero there is at least an element aij 0 , where i j.
Computing the coordinates transformation:
The expression of the quadratic form becomes:
h(x) = aijxi xj
where at least an element aii is non-zero. Any quadratic form can be expressed analytically into an expression, where at least an element of the first diagonal line of the matrix A is non-zero, using a coordinate transformation, meaning a base change in the vector space Vn .
In the following part we will demonstrate, using induction on n, that a quadratic form with at least one non-zero element on the first diagonal line, can be reduced to a canonical form, using successive base change of the vector space Vn .
Without limiting the generality, we assume a11 0 , and in this case we shall write the analytical expression of h as follows:
h(x) =a11 x12 + 2a1k x1 xk + aij xi xj
We add to the analytical expression below the necessary terms in order to have the square of the expression a11 x1 + a12 x2 + + a1nxn , and we obtain:
h(x) = ( a11 x1 + a12 x2 + + a1nxn )2 + . (3.2)
Computing the change of the coordinates:
x1 = a11 x1 + a12 x2 + + a1nxn
xj = xj , j= ,
which is equivalent to a base change in the vector space Vn, the expression of the quadratic form in this base can be written as follows:
h(x ) = x12 + aij xixj (3.3)
The expression (x) = aij xixj from (3.3) is a quadratic form of n 1 variables. Repeating the process described above, after no more than n-1 steps we will obtain the quadratic form h with respect to a base B*, written as the sum of r = rank h n square. This expression represents the canonical quadratic form h . q.e.d.
Let h be a quadratic form over Vn and A=(aij) the associated matrix with respect to a base B Vn .
If all of the main determinants
= a11 , = , , n=det.A
are non-zero then there is a quadratic form h with respect to a base of Vn which admits the canonical expression:
h( x) = 2 , (3.4)
where 0=1 and , i=, are the coordinates of the vector x with respect to the base .
Let Vn be an Euclidian vector space and h : VnRn a quadratic form. In the vector space Vn there is an orthonormed base B and the quadratic form h with respect to the base B has the following canonical expression:
h(x) = i (x i)2 , (3.5)
where l l ln , are eigenvalues of the matrix of the quadratic form h of a given base, and x i are the coordinates of the vector x in the base B
Proof. Into a given base B, the quadratic form h is caracterized by a symmetrical real matrix A. If we denote by l l ln the n eigenvalues (some of them may be equal to one another) of the matrix A, then the coordinates of orthonormed eigenvectors, written as columns, represents the diagonalizing matrix H for the matrix A. In a base formed by eigenvectors, the quadratic form h admits a canonical form given by the relation (3.5), q.e.d.
Let h : VnRn be a quadratic form and its canonical form
h(x) = a1X12 + a2X22 + . . . +arXr2 , r = rank h, (3.6)
obtained by one of the methods presented above.
If we denote by p, the number of strictly positive coefficients from the canonical expression (3.6), named positive index of inertia of h, and we denote by q = r - p the number of strictly negative coefficients (3.6), named negative index of inertia, then the whole number s = p q will be called the signature of the quadratic form h.
(inertia law ) The signature of a quadratic form h is the same in every canonical expression of the quadratic form (the signature doesnt depend on the method used to obtain the canonical expression).
A quadratic form h is called :
a) positively defined if h(x) > xIV
b)negatively defined if h(x) < xIV
c)semi-defined positively if h(x) xIV
negatively if h(x) xIV
and y I V such that h(y) = 0
d)undefined if x, yIV such that h(x) > 0 and h(y)<
From the previous definition we note that a quadratic form is positively (negatively) defined if and only if p = n ( q = n ) .
If the conditions of the Jacobi theorem are accomplished then a quadratic form h is:
positively defined Di > i =
negatively defined (-1)kDk > k = .
Study which of the following applications are billinear forms:
a) f : R2 R2 R , f(x,y) = 2x1y1 3x2y2
b) f : R3 R3 R , f(x,y ) = x1y3 +x2y2 3x3y1
c) f : R3 R3 R , f(x,y ) =x12y2 + x3y1
2. Let g : R3 R3 R be a billinear form defined by:
g(x,y) = x1y2 +x1y3 2x2y2 + 2x3y1
a) The matrix of g in the canonical base of R3
b) The matrix of g in the following base .
c) The rank of g .
3. Let g : R3 R3 R be an application g(x,y) = 2x1y2+2x2y13x3y3 .
a) Show that g is a symmetrical billinear form
b) Find the quadratic form associated to g.
c) Determine the matrix of g with respect to the base B = where e1=(1,1,0), e2=(1,0,1), e3 =(0,1,1).
4. Determine the symmetrical billinear form attached to the quadratic form h : R3 R , h(x) = x12 2x22 + 4x1x3 6x2x3 and find the analytical expression of it with respect to the base B =
Determine the orthogonal complement of the subspace generated by the following vectors: e1 =(2,-1,0), e2 = (0,2,-1) with respect to the quadratic form h : R3 R, h(x) = x12 + x22 2x1x3 .
Using the Gauss method, reduce to a canonical form and determine the corresponding base of the canonical form for the following quadratic form:
a) h : R3 R , h(x) = 2x12 + x22 - 4x1x2 2x1x3
b) h : R3 R , h(x) = x1x2 x1x3 + 2x2x3
Specify the signature of this quadratic form.
Using the Jacobi method determine the canonical form and signature and the base of the canonical form found for:
a) h : R3 R , h(x) = x12 +3 x22 +x32 + 4x1x2 + 6x1x3 + 8x2x3
b) h : R3 R , h(x) = x12 + x22 + x32 x42 + 4x2x3 + 2x2x4 + 6x3x4 .
8. Determine the orthonormed base with respect to the following quadratic form which admits a canonical form:
a) h : R2 R , h(x) = 5x12 +5x22 + 4x1x2
b) h : R3 R , h(x) = 2x1x2 +2x1x3 + 2x2x3
c) h : R4 R , h(x) = x12+x22+x32+x42+2x1x2+2x1x4-2x2x32x2x42x3x4.
Determine the signature of these quadratic forms.
Check the inertia law for the quadratic form h : R3 R ,
h(x) = -x12 +2x1x2 + 2x22 + 6x2x3 .
Determine the real parameter value l for whom the following quadratic forms are positively and respectively negatively defined:
a) h : R2 R , h(x) = x12 +(l + 3)x22 2(l + 1)x1x2
b) h : R3 R , h(x) = x12 + x22 + lx3 2 2x1x2 4x1x3 .
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