# 13. Quadratic forms, polynomials and hypersurfaces

A quadratic form is a function which assigns a number to each vector, in such a way that it is sum of products of two coordinates, e.g. . The square of the norm is also an example of a quadratic form ().

In other words, a quadratic form can be described as , where is a bilinear form. We can always take a symmetric bilinear form, if the characteristic of the field is not equal to , and we are going to make such an assumption from now on.

### Positive and negative definite forms

We can classify forms with respect to possible sign of results:

• form is positively definite, if for all , we get .
• form is negatively definite, if for all , we get .
• form is positively semidefinite, if for all , we get .
• form is negatively semidefinite, if for all , we get .

Obviously a form may not fall in any of those categories, if for some we have and . Such forms are called indefinite.

### Matrix of a form

The matrix of a quadratic form with respect to basis is the matrix , where is a symmetric bilinear form such that . E.g., let , then:

so notice, that the coefficients are divided by outside the diagonal, because the same expression is generated twice.

### Sylvester’s criterion

Sylvester’s criterion determines whether a form is positively definite or negatively definite. Notice that it does not tell anything about the categories with semidefinite forms!

How does it work? We study determinants of minors: let be the matrix of size in the left upper corner of the matrix of a form we study. Let be the size of the matrix of this form. Sylvester’s criterion consists of the two following facts:

• if for any we have , the form is positively definite,
• if for any we have for even , and , for odd , then the form is negatively definite.

E.g. let , so its matrix: , so and , therefore and , so form is positively definite.

E.g., let , so its matrix: , so and , also , therefore and and , so form is negatively definite.

Finally let , its matrix: , so and , therefore and , so form is neither positively definite nor negatively definite.

### Diagonalization of quadratic forms

But to check everything (including semi definiteness), we have to diagonalize the form, i.e. find a basis in which its matrix is diagonal (a diagonal congruent matrix). Then, obviously if:

• it has only positive entrees on the diagonal, then is positive definite,
• it has only negative entrees on the diagonal, then is negative definite,
• it has only nonnegative entrees on the diagonal, then is positive definite,
• it has only nonpositive entrees on the diagonal, then is negative semi definite,
• it has a positive and a negative entree on the diagonal, then is nondefinite.

It can be done it the tree following methods

### Diagonalization of a form: complementing to squares

We may complement a formula of a form to squares making sure to use all expressions with the first variable first, and then all with the second one, and so on.

E.g.

where , i , so the form is non-definite. The basis , in which the formula is expressed is , because

### Diagonalization of a form: orthogobal basis

We may also find an orthogonal basis with respect to the symmetrical bilinear form related to the considered quadratic form. Then the entrees on the diagonal are the values of the form on the vectors from this basis.

### Diagonalization of a form: eigenvalues

Finally, we shall remind ourselves that there exists a basis consisting of eigenvectors of a self-adjoint endomorphism described by the same matrix, which is orthogonal with respect to the symmetrical bilinear form related to the considered quadratic form. Then the entrees on the diagonal are the eigenvalues of the matrix.

E.g.: let , the matrix: , so its characteristic polynomial: has zeroes in and , so it has eigenvalues of both signs, so is is indefinite.

### Polynomials and polynomial functions

A polynomial of degree over field with variables is an expression of form

with at least one for not equal to zero, and the space of all polynomials over with variables is denoted by . E.g. is a polynomial of degree in .

Given an affine -dimensional space and a basic system , a polynomial defines a function, called a polynomial function, given as

which obviously abuses the notation, since denotes the polynomial itself and the polynomial function. But for infinite fields it is not a problem,, because we have a bijection between these two sets.

Notice also, that if can be written as a polynomial function in a basic system, it can written in this form in any basic system. Moreover, the related polynomial is of the same degree.

### Hypersurfaces and algebraic sets

is an algebraic set, if

where are polynomial functions. It is a hypersurface, if

where is a polynomial funkction.

It is easy to notice that over it it the same thing, since we can take .

### Equivalence relation on polynomial functions and hypersurfaces

Two polynomial functions are equivalent, if there exist basic systems and , such that in is described by the same polynomial as in .

Two hypersurfaces are equivalent, if the functions describing them are equivalent. It is so if and only if the second hypersurface is an image of the first one under an affine isomorphism on .

### Canonical form of a polynomial function: hypersurfaces of second degree

Fix a hypersurface of second degree, i.e. described by a polynomial function of second degree. We want to find an equivalent polynomial function of a simplest possible form, i.e. in a canonical form. In other words, we will be looking for a basic system in which the equation describing the hypersurface is as simple as possible.

Every polynomial function of second degree is a sum of a quadratic form and a an affine function, i.e. e.g. if then , where and .

For every polynomial function of second degree (thus every equation describing a hypersurface of second degree) it is possible to find a basic system in which the function takes one of the following forms:

, or

, , where is the rank of .

How to transform the function to such a form? First we have to diagonalize the form and describe the function in the new basis. It changes the basis but not the origin of the basic system. Now, for each variable appearing with a non-zero coefficient in for which we have with a non-zero coefficient , we can introduce a new variable , because then . This changes the constant and the origin from to , where is the basic vector related to the variable . If there are no other variables in , we have already reached the first form. If there are other variables (for which there is no in ), then we make from them and the constant coefficient a new variable , which obviously change both the basis and the origin of the basic system.

where , , so , and for the basic system we get

And so

where and , so the final basic system is .

In the second case, let , and then:

where , , , and the basic system is .

where , and , and since

so the final basic system is: .

### Canonical form of a polynomial function of second degree: hypersurfaces over or

Notice that additionally an equation can be always divided by a free variable (if it is non-zero), changing it into and if we are over , for every expression the basic vector related to can be divided by which changes this expression to (and over even by changing this expression to ).

Thus for every equation of second degree over , there is a basic system in which the equation takes one of the following forms:

, lub

, lub

.

Thus for every equation of second degree over , there is a basic system in which the equation takes one of the following forms:

, lub

, lub

.

E.g. reformulating further on the equation is basic system , we see that it is equivalent to , i.e. with equation , which is in basic system (because ). Meanwhile, over it is equivalent to , in basic system , (because this time ).

### Centres of symmetry

A point is a centre of symmetry of a hypersurface described by equation , if for every we have if and only if . It is easy to prove that is a centre of symmetry if and only if it is a critical point of the function , i.e. its partial derivatives are zero at these points.

Thus, if we consider the canonical forms of the equations, we see that a hypersurface described by

has a centre of symmetry and it is in it if and only if . On the other hand, a hypersurface described by

has no centre of symmetry.

### Affine types of hypersurfaces of second degree over

The above means that the type of equation which describes a given hyperspace is a constant. I.e. if in a basic system it takes one of the forms

, or

, or

, in any other basic system it cannot take any of the other forms.

Moreover, if is the number of variables with coefficient in the above equation, then if the equation is

then is the same for every basic system in which this equation is in the canonical form.
If the equation is of form

, or

, then we also get such a result but up to multiplication by , i.e. in every basic system in which the equation is in the canonical form, we get or variables with coefficient .

These possibilities are called the affine types of hypersurfaces.

We shall say that the hypersurface is proper, if it is not included in any hyperspace.

### Affine types of proper curves of second degree in

Thus we have the following affine types of proper curves of second degree in :

two parallel lines

hyperbola

ellipse

a pair of intersecting lines

parabola

### Affine types of proper surfaces of second degree in

Thus we have the following affine types of proper surfaces of second degree in (images by Wikipedia):

a pair of parallel planes

hyperbolic cylinder

elliptic cylinder

hyperboloid of two sheets

hyperboloid of one sheet

ellipsoid

pair of intersecting planes

elliptic cone

parabolic cylinder

elliptic paraboloid

hyperbolic paraboloid