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BILLINIAR FORMS AND QUADRATIC FORMS
§1. Billinear forms


Let V be a vector space over the field K.
1.1 Definition 
It is called billinear form over the vector space V an application g: V_{}V_{}K, 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) _{} and _{}. 
In other words, a billinear form is an application g : V_{}V_{}K, in both of arguments.
Example 1. Canonical scalar product on the vector space R^{n}^{ }
< , > : R^{n}_{}^{} R^{n}_{}^{} R^{n}, having the analytic expression <x,y> = x_{1}y_{1} + x_{2}y_{2} + x_{n}y_{n} 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.
1.2 Definition 
A billinear form g: V_{}V_{}K 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 V_{n } a ndimensional vector space , B a basis of the vector space V_{n} and two arbitrary vectors x = _{}_{ }and y_{j }= _{}.
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 a_{ij }= g(e_{i},e_{j}), where i,j = 1,2, ,n, the expression (1.1) can be written
g(x,y) _{} (1.2)
called the analytical expression of the billinear form g and the matrix A = (a_{ij}) 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) = ^{t}XAY (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 V_{n}, can be associated with a symmetrical (skew symmetrical) matrix.
1.3 Theorem 
If Ω_{} M_{n} ( K ) is the passing matrix from the basis B to the basis B , in the vector space V_{n } and A, A' are the associated matrices to the billinear g form with respect to the considered bases, then A' = ^{t}ΩAΩ (1.3) 
Proof. The expression of the billinear form g in the basis B' is given by
g(x,y) = ^{t}X^{′}A^{′}Y^{′}. Using the coordinates transformation relations:
X= Ω ^{t}X, Y= Ω ^{t}Y ; passing from the basis B to the basis B , we obtain
g(x,y) = ^{t} XAY =^{t}( X′)A( Y′) =^{ t}X(^{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 nondegenerated .
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 nondegenerated (degenerated) billinear form is justified as being that billinear form g:V_{}V_{}K whos associated matrix A is nondegenerated (degenerated), with respect to a base B from the vector space V.
1.4 Definition. 
Let g : V_{}V_{}K 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.
1.6 Theorem. 
If g: V_{}V_{}K is a symmetrical billinear form, then rank g + dim Ker g = n. (1.5) 
Proof: Consider B a basis of the vector space V_{n} and A=(a_{ij}) the matrix of the billinear form with respect to this basis.
If x = _{}_{i} e_{i} and y = _{}_{j }e_{j}_{ }are any two vectors from V_{n}, then
g(x,y) = _{}_{ij}x_{i}y_{j} = _{}y_{j }.
The relation x_{}Ker g is true if and only if x _{} V_{n} is one of solutions of the linear and homogeneous equations system:
_{}a_{ij} x_{i} = 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).
1.6 Consequence 
The symmetrical billinear form g is no degenerated if and only if Ker g =. 
§2. Quadratic forms
2.1 Definition 
A map h: V_{}K, defined on a K vector space V, with the property that a symmetrical billinear form g:V_{}V_{}K exists, such that:h(x) = g(x,x) , _{}x_{}V . 
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 R^{n} uniquely defines the quadratic form
h(x) =< x,x > = x ^{ } _{}R^{n} ,
which is the square of Euclidian norm.
Let us consider a finite dimensional vector space V_{n}, B one of its bases, and x = _{}_{i} e_{i} an arbitrary vector from V_{n}.
The analytical expression of the quadratic form h is given by
h(x) = g(x,x) = _{}_{ij}x_{i}x_{j} = ^{t}XAX, (2.3)
where A = (a_{ij}), 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 .
2.2 Definition 
The vectors x,y_{}V are called orthogonal to the symmetrical billinear form g (or to the quadratic form h) if g(x,y) = 0 . 
If U_{}V_{n} is a vector subspace of V_{n}, then the set U^{┴} = is a vector subspace of V, called the orthogonal complement of U . Moreover, if g is nondegenerated on the subspace U of the finite dimensional space V_{n} then, U_{}U^{┴} =V_{n} .
The set U_{}V 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,y_{}U, where x_{}y .
If the subset B =_{}V_{n} is an orthogonal base of the vector space V_{n}, with respect to g, then the matrix of the billinear form g is a diagonal matrix. Indeed, a_{ij} = g(e_{i},e_{j}) = 0, _{}i _{}j.
In this case, the analytical expression of the symmetrical billinear form g is:
g(x,y) = _{}a_{ii} x_{i} y_{i }_{ }
and the analytical expression of the quadratic form h is given by
h(x) = _{}a_{ii} x_{i}^{2}
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: V_{n}K a quadratic form on V_{n } and A the symmetrical matrix which represents the quadratic form h with respect to the basis B _{} V_{n}. The analytical expression of the quadratic form h in this basis is:
h(x)= _{}or using matrices h(x) = ^{t}XAX (3.1)
If we change the base of the vector space V_{n}, 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.
3.1 Theorem (Gauss) 
Any expression of a quadratic form over a vector space V_{n }_{ }can be reduced to the canonical form, using a change of base. 
Proof. Let h(x) = _{}_{ij}x_{i}x_{j }be_{ }the analytical expression of a nonzero quadratic form h. First we consider the case a_{ii} = 0 , i = _{}. Because h is not identically zero there is at least an element a_{ij} _{} 0 , where i _{}j.
Computing the coordinates transformation:
_{} (3.1)
The expression of the quadratic form becomes:
h(x) = _{}a^{}_{ij}x^{}_{i }x^{}_{j}
where at least an element a^{}_{ii} is nonzero. 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 nonzero, using a coordinate transformation, meaning a base change in the vector space V_{n} .
In the following part we will demonstrate, using induction on n, that a quadratic form with at least one nonzero element on the first diagonal line, can be reduced to a canonical form, using successive base change of the vector space V_{n} .
Without limiting the generality, we assume a^{}_{11} _{}0 , and in this case we shall write the analytical expression of h as follows:
h(x) =a^{}_{11} x^{}_{1}^{2} + 2_{}a_{1k} x^{}_{1 }x_{k} + _{}a_{ij }x_{i} x^{}_{j}
We add to the analytical expression below the necessary terms in order to have the square of the expression a^{}_{11 }x^{}_{1} + a^{}_{12 }x^{}_{2} + + a^{}_{1n}x^{}_{n , }and we obtain:
h(x) = _{}( a^{}_{11 }x^{}_{1} + a^{}_{12 }x^{}_{2} + + a^{}_{1n}x^{}_{n} )^{2} + _{}. (3.2)
Computing the change of the coordinates:
x_{1} = a^{}_{11 }x^{}_{1} + a^{}_{12 }x^{}_{2} + + a^{}_{1n}x^{}_{n}
x_{j} = x_{j } , j=_{} ,
which is equivalent to a base change in the vector space V_{n}, the expression of the quadratic form in this base can be written as follows:
h(x ) = _{}x_{1}^{2 } + _{}a_{ij} x_{i}^{}x_{j}^{} (3.3)
The expression _{}(x) = _{}a_{ij} x_{i}^{}x_{j}^{} from (3.3) is a quadratic form of n 1 variables. Repeating the process described above, after no more than n1 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.
3.2 Theorem (Jacobi) 
Let h be a quadratic form over V_{n } and A=(a_{ij}) the associated matrix with respect to a base B _{} V_{n} . If all of the main determinants_{} = a_{11} , _{} = _{} , , _{}_{n}=det.A are nonzero then there is a quadratic form h with respect to a base _{} of V_{n }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 _{}. 
3.3 Theorem (eigenvalues method) 
Let V_{n} be an Euclidian vector space and h : V_{n}R^{n} a quadratic form. In the vector space V_{n} 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_{ } l_{n} , 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_{ } l_{n }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 : V_{n}R^{n} be a quadratic form and its canonical form
h(x) = a_{1}X_{1}^{2} + a_{2}X_{2}^{2} + . . . +a_{r}X_{r}^{2} , 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.
3.3 Theorem (Sylvester) 
(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). 
3.4 Definitions 
A quadratic form h is called : a) positively defined if h(x) > xIV b)negatively defined if h(x) < xIV c)semidefined 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)< 
Remark:
From the previous definition we note that a quadratic form is positively (negatively) defined if and only if p = n ( q = n ) .
3.3 Theorem (Sylvesters criterion) 
If the conditions of the Jacobi theorem are accomplished then a quadratic form h is: positively defined D_{i} > i = _{} negatively defined (1)^{k}D_{k} > k = _{} . 
Study which of the following applications are billinear forms:
a) f : R^{2} R^{2} R , f(x,y) = 2x_{1}y_{1} 3x_{2}y_{2}_{ }
b) f : R^{3} R^{3} R , f(x,y ) = x_{1}y_{3} +x_{2}y_{2} 3x_{3}y_{1}
c) f : R^{3} R^{3} R , f(x,y ) =x_{1}^{2}y_{2} + x_{3}y_{1}_{ }
2. Let g : R^{3} R^{3} R be a billinear form defined by:
g(x,y) = x_{1}y_{2} +x_{1}y_{3} 2x_{2}y_{2} + 2x_{3}y_{1}
Determine :
a) The matrix of g in the canonical base of R^{3}
b) The matrix of g in the following base .
c) The rank of g .
3. Let g : R^{3} R^{3} R be an application g(x,y) = 2x_{1}y_{2}+2x_{2}y_{1}3x_{3}y_{3} .
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 e_{1}=(1,1,0), e_{2}=(1,0,1), e_{3} =(0,1,1).
4. Determine the symmetrical billinear form attached to the quadratic form h : R^{3} R , h(x) = x_{1}^{2} 2x_{2}^{2} + 4x_{1}x_{3} 6x_{2}x_{3 }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: e_{1} =(2,1,0), e_{2} = (0,2,1) with respect to the quadratic form h : R^{3} R, h(x) = x_{1}^{2} + x_{2}^{2} 2x_{1}x_{3} .
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 : R^{3} R , h(x) = 2x_{1}^{2} + x_{2}^{2}  4x_{1}x_{2} 2x_{1}x_{3 }
b) h : R^{3} R , h(x) = x_{1}x_{2} x_{1}x_{3} + 2x_{2}x_{3}
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 : R^{3} R , h(x) = x_{1}^{2} +3 x_{2}^{2} +x_{3}^{2} + 4x_{1}x_{2 }+ 6x_{1}x_{3} + 8x_{2}x_{3 }_{ }
_{ }b) h : R^{3} R , h(x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} x_{4}^{2} + 4x_{2}x_{3 }+ 2x_{2}x_{4} + 6x_{3}x_{4} .
8. Determine the orthonormed base with respect to the following quadratic form which admits a canonical form:
a) h : R^{2} R , h(x) = 5x_{1}^{2} +5x_{2}^{2} + 4x_{1}x_{2} _{ }
b) h : R^{3} R , h(x) = 2x_{1}x_{2} +2x_{1}x_{3} + 2x_{2}x_{3 }_{ }
c) h : R^{4} R , h(x) = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+2x_{1}x_{2}+2x_{1}x_{4}2x_{2}x_{3}2x_{2}x_{4}2x_{3}x_{4.}
Determine the signature of these quadratic forms.
Check the inertia law for the quadratic form h : R^{3} R ,
h(x) = x_{1}^{2} +2x_{1}x_{2} + 2x_{2}^{2} + 6x_{2}x_{3} .
Determine the real parameter value l for whom the following quadratic forms are positively and respectively negatively defined:
a) h : R^{2} R , h(x) = x_{1}^{2} +(l + 3)x_{2}^{2} 2(l + 1)x_{1}x_{2} _{ }
b)_{ }h : R^{3} R , h(x) = x_{1}^{2} + x_{2}^{2} + lx_{3 }^{2} 2x_{1}x_{2} 4x_{1}x_{3 }._{ }
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