BILLINIAR FORMS AND QUADRATIC FORMS

1. Billinear forms

Let V be a vector space over the field K.

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 + x_ny_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.

Let be V_n a n-dimensional 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) = ^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 V_n, can be associated with a symmetrical (skew symmetrical) matrix.

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.

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.

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) = _ijx_iy_j = y_j .

The relation xKer 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).

2. Quadratic forms

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 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) = _ijx_ix_j = ^tXAX, (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 .

If UV_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 non-degenerated on the subspace U of the finite dimensional space V_n then, UU^┴ =V_n .

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 =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_i²

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_nK 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) = ^tXAX (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.

Proof. Let h(x) = _ijx_ix_j be the analytical expression of a non-zero 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_ijx_i x_j

where at least an element a_ii 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 V_n .

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 V_n .

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 + 2a_1k x₁ 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 a₁₁ x₁ + a₁₂ x₂ + + a_1nx_{n ,} and we obtain:

h(x) = ( a₁₁ x₁ + a₁₂ x₂ + + a_1nx_n )² + . (3.2)

Computing the change of the coordinates:

x₁ = a₁₁ x₁ + a₁₂ x₂ + + a_1nx_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 ) = x₁² + a_ij x_ix_j (3.3)

The expression (x) = a_ij x_ix_j 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.

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_nRⁿ be a quadratic form and its canonical form

h(x) = a₁X₁² + a₂X₂² + . . . +a_rX_r² , 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.

Remark:

From the previous definition we note that a quadratic form is positively (negatively) defined if and only if p = n ( q = n ) .

Exercises

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

Determine :

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+2x₁x₄-2x₂x₃2x₂x₄2x₃x_4.

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 .