Quadratic shapes.
Sign definiteness of forms. Sylvester criterion

The adjective “quadratic” immediately suggests that something here is connected with a square (the second degree), and very soon we will find out this “something” and what the shape is. It turned out to be a tongue twister :)

Welcome to my new lesson, and as an immediate warm-up we'll look at the striped shape linear. Linear form variables called homogeneous 1st degree polynomial:

- some specific numbers* (we assume that at least one of them is non-zero), a are variables that can take arbitrary values.

* Within the framework of this topic we will only consider real numbers .

We have already encountered the term “homogeneous” in the lesson about homogeneous systems of linear equations, and in in this case it implies that the polynomial does not have a plus constant.

For example: – linear form of two variables

Now the shape is quadratic. Quadratic shape variables called homogeneous polynomial of 2nd degree, each term of which contains either the square of the variable or doubles product of variables. So, for example, the quadratic form of two variables has next view:

Attention! This is a standard entry and there is no need to change anything about it! Despite the “scary” appearance, everything is simple here - double subscripts of constants signal which variables are included in which term:
– this term contains the product and (square);
- here is the work;
- and here is the work.

- I’ll warn you right away gross mistake when they lose the “minus” of the coefficient, not understanding that it refers to a term:

Sometimes there is a “school” design option in the spirit, but only sometimes. By the way, note that the constants don’t tell us anything at all here, and therefore it’s more difficult to remember the “easy notation”. Especially when there are more variables.

And quadratic form of three variables already contains six members:

...why are “two” factors placed in “mixed” terms? This is convenient, and it will soon become clear why.

However general formula Let’s write it down, it’s convenient to arrange it as a “sheet”:

– we carefully study each line – there’s nothing wrong with that!

The quadratic form contains terms with the squares of the variables and terms with their paired products (cm. combinatorial combination formula) . Nothing more - no “lonely X” and no added constant (then you will get not a quadratic form, but heterogeneous polynomial of 2nd degree).

Matrix notation of quadratic form

Depending on the values, the form in question can take on both positive and negative values, and the same applies to any linear form - if at least one of its coefficients is different from zero, then it can be either positive or negative (depending on the values).

This form is called alternating sign. And if everything is transparent with the linear form, then with the quadratic form things are much more interesting:

It is absolutely clear that this form can take on the meaning of any sign, thus the quadratic form can also be alternating.

It may not be:

– always, unless simultaneously equal to zero.

- for anyone vector except zero.

And generally speaking, if for anyone non-zero vector , , then the quadratic form is called positive definite; if so then negative definite.

And everything would be fine, but the definiteness of the quadratic form is visible only in simple examples, and this visibility is lost even with a slight complication:
– ?

One might assume that the form is positively defined, but is this really so? What if there are values ​​at which it is less than zero?

There is a theorem: If everyone eigenvalues matrices of quadratic form are positive * , then it is positive definite. If all are negative, then negative.

* It has been proven in theory that all eigenvalues ​​of a real symmetric matrix valid

Let's write the matrix of the above form:
and from Eq. let's find her eigenvalues:

Let's solve the good old quadratic equation:

, which means the form is defined positively, i.e. for any non-zero values ​​it Above zero.

The considered method seems to work, but there is one big BUT. Already for a three-by-three matrix, looking for proper numbers is a long and unpleasant task; with a high probability you will get a polynomial of the 3rd degree with irrational roots.

What should I do? There is an easier way!

Sylvester criterion

No, not Sylvester Stallone :) First, let me remind you what it is corner minors matrices. This qualifiers which “grow” from its upper left corner:

and the last one is exactly equal to the determinant of the matrix.

Now, actually, criterion:

1) Quadratic form is defined positively if and only if ALL its angular minors are greater than zero: .

2) Quadratic form is defined negative if and only if its angular minors alternate in sign, with the 1st minor being less than zero: , , if – even or , if – odd.

If at least one corner minor opposite sign, then the form alternating sign. If the angular minors are of “that” sign, but there are zero ones among them, then this is a special case, which I will discuss a little later, after we click on more common examples.

Let's analyze the angular minors of the matrix :

And this immediately tells us that the form is not negatively defined.

Conclusion: all corner minors are greater than zero, which means the form is defined positively.

There is a difference with the method eigenvalues? ;)

Let us write the form matrix from Example 1:

the first is its angular minor, and the second , from which it follows that the shape is alternating in sign, i.e. depending on the values, it can take both positive and negative values. However, this is already obvious.

Let's take the form and its matrix from Example 2:

There’s no way to figure this out without insight. But with Sylvester’s criterion we don’t care:
, therefore, the form is definitely not negative.

, and definitely not positive (since all angular minors must be positive).

Conclusion: the shape is alternating.

Warm-up examples for independent decision:

Example 4

Investigate quadratic forms for sign definiteness

A)

In these examples everything is smooth (see the end of the lesson), but in fact, to complete such a task Sylvester's criterion may not be sufficient.

The point is that there are “edge” cases, namely: if for any non-zero vector, then the shape is determined non-negative, if – then negative. These forms have non-zero vectors for which .

Here you can quote the following “accordion”:

Highlighting perfect square, we see right away non-negativity form: , and it is equal to zero for any vector with equal coordinates, for example: .

"Mirror" example negative a certain form:

and an even more trivial example:
– here the form is equal to zero for any vector , where is an arbitrary number.

How to identify non-negative or non-positive forms?

For this we need the concept major minors matrices. A major minor is a minor composed of elements that stand at the intersection of rows and columns with the same numbers. Thus, the matrix has two main minors of the 1st order:
(the element is located at the intersection of the 1st row and 1st column);
(the element is at the intersection of the 2nd row and 2nd column),

and one major minor of 2nd order:
– composed of elements of the 1st, 2nd row and 1st, 2nd column.

The matrix is ​​“three by three” There are seven main minors, and here you’ll have to flex your biceps:
– three minors of the 1st order,
three 2nd order minors:
– composed of elements of the 1st, 2nd row and 1st, 2nd column;
– composed of elements of the 1st, 3rd row and 1st, 3rd column;
– composed of elements of the 2nd, 3rd row and 2nd, 3rd column,
and one 3rd order minor:
– composed of elements of the 1st, 2nd, 3rd row and 1st, 2nd and 3rd column.
Exercise for understanding: write down all the major minors of the matrix .
We check at the end of the lesson and continue.

Schwarzenegger criterion:

1) Non-zero* quadratic form defined non-negative if and only if ALL of its major minors non-negative(greater than or equal to zero).

* The zero (degenerate) quadratic form has all coefficients equal to zero.

2) Non-zero quadratic form with matrix is ​​defined negative if and only if:
– major minors of the 1st order non-positive(less than or equal to zero);
– major minors of the 2nd order non-negative;
– major minors of the 3rd order non-positive(alternation began);
…
– major minor of the th order non-positive, if – odd or non-negative, if – even.

If at least one minor is of opposite sign, then the form is sign-alternating.

Let's see how the criterion works in the above examples:

Let's create a shape matrix, and Firstly Let's calculate the angular minors - what if it is defined positively or negatively?

The obtained values ​​do not satisfy the Sylvester criterion, but the second minor not negative, and this makes it necessary to check the 2nd criterion (in the case of the 2nd criterion will not be fulfilled automatically, i.e. the conclusion is immediately drawn about the sign alternation of the form).

Main minors of the 1st order:
– positive,
major minor of 2nd order:
– not negative.

Thus, ALL major minors are not negative, which means the form non-negative.

Let's write the shape matrix , for which the Sylvester criterion is obviously not satisfied. But we also did not receive opposite signs (since both angular minors are equal to zero). Therefore, we check the fulfillment of the non-negativity/non-positivity criterion. Main minors of the 1st order:
– not positive,
major minor of 2nd order:
– not negative.

Thus, according to Schwarzenegger’s criterion (point 2), the form is non-positively defined.

Now let’s take a closer look at a more interesting problem:

Example 5

Examine the quadratic form for sign definiteness

This uniform is decorated with the “alpha” order, which can be equal to any real number. But it will only be more fun we decide.

First, let’s write down the form matrix; many people have probably already gotten used to doing this orally: on main diagonal We put the coefficients for the squares, and in the symmetrical places we put half the coefficients of the corresponding “mixed” products:

Let's calculate the angular minors:

I will expand the third determinant on the 3rd line:

A homogeneous polynomial of degree 2 in several variables is called a quadratic form.

The quadratic form of variables consists of terms of two types: squares of variables and their pairwise products with certain coefficients. The quadratic form is usually written as the following square diagram:

Pairs of similar terms are written with equal coefficients, so that each of them constitutes half the coefficient of the corresponding product of the variables. Thus, each quadratic form is naturally associated with its coefficient matrix, which is symmetric.

It is convenient to represent the quadratic form in the following matrix notation. Let us denote by X a column of variables through X - a row, i.e., a matrix transposed with X. Then

Quadratic forms are found in many branches of mathematics and its applications.

In number theory and crystallography, quadratic forms are considered under the assumption that the variables take only integer values. In analytical geometry, the quadratic form is part of the equation of a curve (or surface) of order. In mechanics and physics, the quadratic form appears to express kinetic energy systems through the components of generalized velocities, etc. But, in addition, the study of quadratic forms is also necessary in analysis when studying functions of many variables, in questions for the solution of which it is important to find out how this function in the vicinity of a given point deviates from the one approaching it linear function. An example of a problem of this type is the study of a function for its maximum and minimum.

Consider, for example, the problem of studying the maximum and minimum for a function of two variables that has continuous partial derivatives up to order. A necessary condition In order for a point to give a maximum or minimum of a function, the partial derivatives of the order at the point are equal to zero. Let us assume that this condition is met. Let's give the variables x and y small increments and k and consider the corresponding increment of the function. According to Taylor's formula, this increment, up to small higher orders, is equal to the quadratic form where are the values ​​of the second derivatives calculated at the point If this quadratic form is positive for all values ​​of and k (except ), then the function has a minimum at the point; if it is negative, then it has a maximum. Finally, if a form takes both positive and negative values, then there will be no maximum or minimum. Functions of more variables.

The study of quadratic forms mainly consists of studying the problem of equivalence of forms with respect to one or another set of linear transformations of variables. Two quadratic forms are said to be equivalent if one of them can be converted into the other by one of the transformations of a given set. Closely related to the problem of equivalence is the problem of reducing the form, i.e. transforming it to some possibly simplest form.

In various questions related to quadratic forms, various sets of admissible transformations of variables are also considered.

In questions of analysis, any non-special transformations of variables are used; for the purposes of analytical geometry greatest interest present orthogonal transformations, i.e. those that correspond to a transition from one system of variables Cartesian coordinates to another. Finally, in number theory and crystallography linear transformations with integer coefficients and with a determinant equal to unity are considered.

We will consider two of these problems: the question of reducing a quadratic form to its simplest form through any non-singular transformations and the same question for orthogonal transformations. First of all, let's find out how a matrix of quadratic form is transformed during a linear transformation of variables.

Let , where A is a symmetric matrix of form coefficients, X is a column of variables.

Let's do it linear transformation variables, writing it abbreviated as . Here C denotes the matrix of coefficients of this transformation, X is a column of new variables. Then and therefore, so the matrix of the transformed quadratic form is

The matrix automatically turns out to be symmetric, which is easy to check. Thus, the problem of reducing a quadratic form to the simplest form is equivalent to the problem of reducing a symmetric matrix to the simplest form by multiplying it on the left and right by mutually transposed matrices.

The concept of quadratic form. Matrix of quadratic form. Canonical form of quadratic form. Lagrange method. Normal view of a quadratic form. Rank, index and signature of quadratic form. Positive definite quadratic form. Quadrics.

Concept of quadratic form: a function on a vector space defined by a homogeneous polynomial of the second degree in the coordinates of the vector.

Quadratic form from n unknown is called a sum, each term of which is either the square of one of these unknowns, or the product of two different unknowns.

Quadratic matrix: The matrix is ​​called a matrix of quadratic form in a given basis. If the field characteristic is not equal to 2, we can assume that the matrix of quadratic form is symmetric, that is.

Write a matrix of quadratic form:

Hence,

In vector matrix form, the quadratic form is:

A, where

Canonical form of quadratic form: A quadratic form is called canonical if all i.e.

Any quadratic form can be reduced to canonical form using linear transformations. In practice, the following methods are usually used.

Lagrange method : sequential selection of complete squares. For example, if

Then a similar procedure is performed with the quadratic form etc. If in quadratic form everything is but then after preliminary transformation the matter comes down to the procedure considered. So, if, for example, then we assume

Normal form of quadratic form: A normal quadratic form is a canonical quadratic form in which all coefficients are equal to +1 or -1.

Rank, index and signature of quadratic form: Rank of quadratic form A is called the rank of the matrix A. The rank of a quadratic form does not change under non-degenerate transformations of unknowns.

The number of negative coefficients is called the negative form index.

Number positive members in canonical form is called the positive index of inertia of the quadratic form, the number of negative terms is called the negative index. The difference between the positive and negative indices is called the signature of the quadratic form

Positive definite quadratic form: Real quadratic form is called positive definite (negative definite) if, for any real values ​​of the variables that are not simultaneously zero,

. (36)

In this case, the matrix is ​​also called positive definite (negative definite).

The class of positive definite (negative definite) forms is part of the class of non-negative (resp. non-positive) forms.

Quadrics: Quadric - n-dimensional hypersurface in n+1-dimensional space, defined as the set of zeros of a polynomial of the second degree. If you enter the coordinates ( x 1 , x 2 , x n+1 ) (in Euclidean or affine space), general equation quadrics has the form

This equation can be rewritten more compactly in matrix notation:

where x = ( x 1 , x 2 , x n+1 ) — row vector, x T is a transposed vector, Q— size matrix ( n+1)×( n+1) (it is assumed that at least one of its elements is non-zero), P is a row vector, and R— constant. Quadrics over real ones are most often considered complex numbers. The definition can be extended to quadrics in projective space, see below.

More generally, the set of zeros of a system of polynomial equations is known as an algebraic variety. Thus, a quadric is a (affine or projective) algebraic variety of the second degree and codimension 1.

Transformations of plane and space.

Definition of plane transformation. Motion detection. properties of movement. Two types of movements: movement of the first kind and movement of the second kind. Examples of movements. Analytical expression of motion. Classification of plane movements (depending on the presence fixed points and invariant lines). Group of plane movements.

Definition of plane transformation: Definition. A plane transformation that preserves the distance between points is called movement(or movement) of the plane. The plane transformation is called affine, if it transforms any three points lying on the same line into three points also lying on the same line and at the same time preserving the simple relation of the three points.

Motion Definition: These are shape transformations that preserve the distances between points. If two figures are precisely aligned with each other through movement, then these figures are the same, equal.

Movement properties: Every orientation-preserving motion of a plane is either a parallel translation or a rotation; every orientation-changing motion of a plane is either an axial symmetry or a sliding symmetry. When moving, points lying on a straight line transform into points lying on a straight line, and their order is preserved relative position. When moving, the angles between half-lines are preserved.

Two types of movements: movement of the first kind and movement of the second kind: Movements of the first kind are those movements that preserve the orientation of the bases of a certain figure. They can be realized by continuous movements.

Movements of the second kind are those movements that change the orientation of the bases to the opposite. They cannot be realized by continuous movements.

Examples of movements of the first kind are translation and rotation around a straight line, and movements of the second kind are central and mirror symmetries.

The composition of any number of movements of the first kind is a movement of the first kind.

The composition of an even number of movements of the second kind is movement of the 1st kind, and the composition of an odd number of movements of the 2nd kind is movement of the 2nd kind.

Examples of movements:Parallel transfer. Let a be the given vector. Parallel transfer to vector a is a mapping of the plane onto itself, in which each point M is mapped to point M 1, which is the vector MM 1 equal to the vector A.

Parallel translation is a movement because it is a mapping of the plane onto itself, preserving distances. This movement can be visually represented as a shift of the entire plane in the direction given vector but on its length.

Rotate. Let us denote the point O on the plane ( turning center) and set the angle α ( angle of rotation). Rotation of the plane around the point O by an angle α is the mapping of the plane onto itself, in which each point M is mapped to the point M 1, such that OM = OM 1 and the angle MOM 1 is equal to α. In this case, point O remains in its place, i.e., it is mapped onto itself, and all other points rotate around point O in the same direction - clockwise or counterclockwise (the figure shows a counterclockwise rotation).

Rotation is a movement because it represents a mapping of the plane onto itself, in which distances are preserved.

Analytical expression of movement: the analytical connection between the coordinates of the preimage and the image of the point has the form (1).

Classification of plane movements (depending on the presence of fixed points and invariant lines): Definition:

A point on a plane is invariant (fixed) if, under a given transformation, it transforms into itself.

Example: With central symmetry, the point of the center of symmetry is invariant. When turning, the point of the center of rotation is invariant. With axial symmetry, the invariant line is a straight line - the axis of symmetry is a straight line of invariant points.

Theorem: If a movement does not have a single invariant point, then it has at least one invariant direction.

Example: Parallel transfer. Indeed, straight lines parallel to this direction are invariant as a figure as a whole, although it does not consist of invariant points.

Theorem: If some ray moves, the ray translates into itself, then this movement is either identity transformation, or symmetry with respect to the straight line containing the given ray.

Therefore, based on the presence of invariant points or figures, it is possible to classify movements.

Movement name	Invariant points	Invariant lines
Movement of the first kind.
1. - turn	(center) - 0	No
2. Identical transformation	all points of the plane	all straight
3. Central symmetry	point 0 - center	all lines passing through point 0
4. Parallel transfer	No	all straight
Movement of the second kind.
5. Axial symmetry.	set of points	axis of symmetry (straight line) all straight lines

Plane motion group: In geometry important role groups of self-combining figures play. If is a certain figure on a plane (or in space), then we can consider the set of all those movements of the plane (or space) during which the figure turns into itself.

This set is a group. For example, for equilateral triangle the group of movements of the plane that transform the triangle into itself consists of 6 elements: rotations by angles around a point and symmetries about three straight lines.

They are shown in Fig. 1 red lines. Elements of a self-combination group regular triangle can be specified differently. To explain this, let us number the vertices of a regular triangle with the numbers 1, 2, 3. Any self-alignment of the triangle takes points 1, 2, 3 to the same points, but taken in a different order, i.e. can be conditionally written in the form of one of these brackets:

etc.

where the numbers 1, 2, 3 indicate the numbers of those vertices into which vertices 1, 2, 3 go as a result of the movement under consideration.

Projective spaces and their models.

The concept of projective space and the model of projective space. Basic facts of projective geometry. A bunch of lines centered at the point O is a model of the projective plane. Projective points. The extended plane is a model of the projective plane. Extended three-dimensional affine or Euclidean space is a model of projective space. Images of flat and spatial figures in parallel design.

The concept of projective space and the model of projective space:

Projective space over a field is a space consisting of lines (one-dimensional subspaces) of some linear space over a given field. Direct spaces are called dots projective space. This definition can be generalized to an arbitrary body

If it has dimension , then the dimension of the projective space is called number , and the projective space itself is denoted and called associated with (to indicate this, the notation is adopted).

Transfer from vector space dimension to the corresponding projective space is called projectivization space.

Points can be described using homogeneous coordinates.

Basic facts of projective geometry: Projective geometry is a branch of geometry that studies projective planes and spaces. main feature Projective geometry is based on the principle of duality, which adds graceful symmetry to many designs. Projective geometry can be studied both purely geometric point point of view, both analytical (using homogeneous coordinates) and salgebraic, considering the projective plane as a structure over a field. Often, and historically, the real projective plane is considered to be the Euclidean plane with the addition of "line at infinity".

Whereas the properties of figures with which Euclidean geometry deals are metric (specific quantities angles, segments, areas), and the equivalence of figures is equivalent to their congruence(i.e. when figures can be translated into one another through movement while preserving metric properties), there are more “deep-lying” properties geometric shapes, which are preserved during transformations of more than general type than movement. Projective geometry deals with the study of properties of figures that are invariant under the class projective transformations, as well as these transformations themselves.

Projective geometry complements Euclidean, providing beautiful and simple solutions for many problems complicated by the presence of parallel lines. Especially simple and elegant projective theory conic sections.

There are three main approaches to projective geometry: independent axiomatization, complementation of Euclidean geometry, and structure over a field.

Axiomatization

Projective space can be defined using different set axiom.

Coxeter provides the following:

1. There is a straight line and a point not on it.

2. Each line has at least three points.

3. Through two points you can draw exactly one straight line.

4. If A, B, C, And D — various points And AB And CD intersect, then A.C. And BD intersect.

5. If ABC is a plane, then there is at least one point not in the plane ABC.

6. Two different planes intersect in at least two points.

7. The three diagonal points of a complete quadrilateral are not collinear.

8. If three points are on a line X X

The projective plane (without the third dimension) is defined by slightly different axioms:

1. Through two points you can draw exactly one straight line.

2. Any two lines intersect.

3. There are four points, of which three are not collinear.

4. The three diagonal points of complete quadrilaterals are not collinear.

5. If three points are on a line X are invariant with respect to the projectivity of φ, then all points on X invariant with respect to φ.

6. Desargues' theorem: If two triangles are perspective through a point, then they are perspective through a line.

In the presence of a third dimension, Desargues' theorem can be proven without introducing an ideal point and line.

Extended plane - projective plane model: In the affine space A3 we take a bundle of lines S(O) with center at the point O and a plane Π that does not pass through the center of the bundle: O 6∈ Π. A bundle of lines in an affine space is a model of the projective plane. Let's define a mapping of the set of points of the plane Π onto the set of straight lines of the connective S (Fuck, pray if you got this question, forgive me)

Extended three-dimensional affine or Euclidean space—a model of projective space:

In order to make the mapping surjective, we repeat the process of formally extending the affine plane Π to the projective plane, Π, supplementing the plane Π with a set of improper points (M∞) such that: ((M∞)) = P0(O). Since in the map the inverse image of each plane of the bundle of planes S(O) is a line on the plane d, it is obvious that the set of all improper points of the extended plane: Π = Π ∩ (M∞), (M∞), represents an improper line d∞ of the extended plane, which is the inverse image of the singular plane Π0: (d∞) = P0(O) (= Π0). (I.23) Let us agree that here and henceforth we will understand the last equality P0(O) = Π0 in the sense of equality of sets of points, but endowed with a different structure. Adding affine plane improper line, we have achieved that the mapping (I.21) has become bijective on the set of all points of the extended plane:

Images of flat and spatial figures during parallel design:

In stereometry, spatial figures are studied, but in the drawing they are depicted as flat figures. How should a spatial figure be depicted on a plane? Typically in geometry, parallel design is used for this. Let p be some plane, l- a straight line intersecting it (Fig. 1). Through arbitrary point A, not belonging to the line l, draw a line parallel to the line l. The point of intersection of this line with the plane p is called the parallel projection of the point A to the plane p in the direction of the straight line l. Let's denote it A". If the point A belongs to the line l, then by parallel projection A the point of intersection of the line is considered to be on the plane p l with plane p.

Thus, each point A space its projection is compared A" onto the plane p. This correspondence is called parallel projection onto the plane p in the direction of the straight line l.

Group of projective transformations. Application to problem solving.

The concept of projective transformation of a plane. Examples of projective transformations of the plane. Properties of projective transformations. Homology, properties of homology. Group of projective transformations.

The concept of projective transformation of a plane: The concept of a projective transformation generalizes the concept of a central projection. If you do central projection plane α onto some plane α 1, then the projection of α 1 onto α 2, α 2 onto α 3, ... and finally, some plane α n again on α 1, then the composition of all these projections is the projective transformation of the plane α; Parallel projections can also be included in such a chain.

Examples of projective plane transformations: A projective transformation of a completed plane is its one-to-one mapping onto itself, in which the collinearity of points is preserved, or, in other words, the image of any line is a straight line. Every projective transformation is a composition of a chain of central and parallel projections. An affine transformation is special case projective, in which the infinitely distant straight line turns into itself.

Properties of projective transformations:

During a projective transformation, three points not lying on a line are transformed into three points not lying on a line.

During a projective transformation, the frame turns into a frame.

During a projective transformation, a line goes into a straight line, and a pencil goes into a pencil.

Homology, properties of homology:

A projective transformation of a plane that has a line of invariant points, and therefore a pencil of invariant lines, is called homology.

1. A line passing through non-coinciding corresponding homology points is an invariant line;

2. Lines passing through non-coinciding corresponding homology points belong to the same pencil, the center of which is an invariant point.

3. The point, its image and the center of homology lie on the same straight line.

Group of projective transformations: consider the projective mapping of the projective plane P 2 onto itself, that is, the projective transformation of this plane (P 2 ’ = P 2).

As before, the composition f of projective transformations f 1 and f 2 of the projective plane P 2 is the result of sequential execution of transformations f 1 and f 2: f = f 2 °f 1 .

Theorem 1: the set H of all projective transformations of the projective plane P 2 is a group with respect to the composition of projective transformations.

Positive definite quadratic forms

Definition. Quadratic form from n unknowns are called positive definite, if its rank is equal to the positive inertia index and equal to the number unknown.

Theorem. A quadratic form is positive definite if and only if, on any nonzero set of variable values, it takes positive values.

Proof. Let the quadratic form be a non-degenerate linear transformation of the unknowns

brought back to normal

.

For any non-zero set of variable values, at least one of the numbers different from zero, i.e. . The necessity of the theorem is proven.

Suppose that the quadratic form takes positive values ​​on any non-zero set of variables, but its positive inertia index is a non-degenerate linear transformation of the unknowns

Let's bring it to normal form. Without loss of generality, we can assume that in this normal form the square of the last variable is either absent or included with a minus sign, i.e. , where or . Let us assume that is a non-zero set of variable values ​​obtained as a result of solving the system linear equations

In this system, the number of equations is equal to the number of variables and the determinant of the system is nonzero. According to Cramer's theorem, the system has a unique solution, and it is non-zero. For this set. Contradiction with the condition. We come to a contradiction with the assumption, which proves the sufficiency of the theorem.

Using this criterion, it is impossible to determine from the coefficients whether the quadratic form is positive definite. The answer to this question is given by another theorem, for the formulation of which we introduce another concept. Principal diagonal minors of a matrix are minors located in its left top corner:

, , , … , .

Theorem.A quadratic form is positive definite if and only if all its principal diagonal minors are positive.

Proof we will carry out the method of complete mathematical induction by number n quadratic variables f.

Induction hypothesis. Let us assume that for quadratic forms with fewer variables n the statement is true.

Consider the quadratic form of n variables. Let's put all the terms containing . The remaining terms form a quadratic form of the variables. According to the induction hypothesis, the statement is true for her.

Let us assume that the quadratic form is positive definite. Then the quadratic form is positive definite. If we assume that this is not the case, then there is a non-zero set of variable values , for which and correspondingly, , and this contradicts the fact that the quadratic form is positive definite. By the induction hypothesis, all principal diagonal minors of a quadratic form are positive, i.e. all first principal minors of quadratic form f are positive. Last principal minor of quadratic form – this is the determinant of its matrix. This determinant is positive, since its sign coincides with the sign of the matrix of its normal form, i.e. with the sign of the determinant of the identity matrix.

Let all the principal diagonal minors of the quadratic form be positive. Then all the principal diagonal minors of the quadratic form are positive from the equality . By the induction hypothesis, the quadratic form is positive definite, so there is a non-degenerate linear transformation of the variables that reduces the form to the form of the sum of squares of the new variables. This linear transformation can be extended to a non-degenerate linear transformation of all variables by setting . This transformation reduces the quadratic form to the form

Quadratic shapes

Quadratic shape f(x 1, x 2,...,x n) of n variables is a sum, each term of which is either the square of one of the variables, or the product of two different variables, taken with a certain coefficient: f(x 1, x 2, ...,x n) = (a ij = a ji).

The matrix A composed of these coefficients is called a matrix of quadratic form. It's always symmetrical matrix (i.e. a matrix symmetrical about the main diagonal, a ij = a ji).

In matrix notation, the quadratic form is f(X) = X T AX, where

Indeed

For example, let's write in matrix form quadratic form.

To do this, we find a matrix of quadratic form. Its diagonal elements are equal to the coefficients of the squared variables, and the remaining elements are equal to the halves of the corresponding coefficients of the quadratic form. That's why

Let the matrix-column of variables X be obtained by a non-degenerate linear transformation of the matrix-column Y, i.e. X = CY, where C - non-singular matrix nth order. Then the quadratic form
f(X) = X T AX = (CY) T A(CY) = (Y T C T)A(CY) = Y T (C T AC)Y.

Thus, with a non-degenerate linear transformation C, the matrix of quadratic form takes the form: A * = C T AC.

For example, let's find the quadratic form f(y 1, y 2), obtained from the quadratic form f(x 1, x 2) = 2x 1 2 + 4x 1 x 2 - 3x 2 2 by linear transformation.

The quadratic form is called canonical(It has canonical view), if all its coefficients a ij = 0 for i ≠ j, i.e.
f(x 1, x 2,...,x n) = a 11 x 1 2 + a 22 x 2 2 + … + a nn x n 2 = .

Its matrix is ​​diagonal.

Theorem(proof not given here). Any quadratic form can be reduced to canonical form using a non-degenerate linear transformation.

For example, let us reduce the quadratic form to canonical form
f(x 1, x 2, x 3) = 2x 1 2 + 4x 1 x 2 - 3x 2 2 – x 2 x 3.

To do this, we first select perfect square with variable x 1:

f(x 1, x 2, x 3) = 2(x 1 2 + 2x 1 x 2 + x 2 2) - 2x 2 2 - 3x 2 2 – x 2 x 3 = 2(x 1 + x 2) 2 - 5x 2 2 – x 2 x 3.

Now we select a complete square with the variable x 2:

f(x 1, x 2, x 3) = 2(x 1 + x 2) 2 – 5(x 2 2 – 2* x 2 *(1/10)x 3 + (1/100)x 3 2) - (5/100)x 3 2 =
= 2(x 1 + x 2) 2 – 5(x 2 – (1/10)x 3) 2 - (1/20)x 3 2.

Then the non-degenerate linear transformation y 1 = x 1 + x 2, y 2 = x 2 – (1/10) x 3 and y 3 = x 3 brings this quadratic form to the canonical form f(y 1, y 2, y 3) = 2y 1 2 - 5y 2 2 - (1/20)y 3 2 .

Note that the canonical form of a quadratic form is determined ambiguously (the same quadratic form can be reduced to the canonical form different ways). However, the received different ways canonical forms have a number of general properties. In particular, the number of terms with positive (negative) coefficients of a quadratic form does not depend on the method of reducing the form to this form (for example, in the example considered there will always be two negative and one positive coefficient). This property is called law of inertia of quadratic forms.

Let us verify this by bringing the same quadratic form to its canonical form in a different way. Let's start the transformation with the variable x 2:
f(x 1, x 2, x 3) = 2x 1 2 + 4x 1 x 2 - 3x 2 2 – x 2 x 3 = -3x 2 2 – x 2 x 3 + 4x 1 x 2 + 2x 1 2 = - 3(x 2 2 –
- 2* x 2 ((1/6) x 3 + (2/3)x 1) +((1/6) x 3 + (2/3)x 1) 2) – 3((1/6) x 3 + (2/3)x 1) 2 + 2x 1 2 =
= -3(x 2 – (1/6) x 3 - (2/3)x 1) 2 – 3((1/6) x 3 + (2/3)x 1) 2 + 2x 1 2 = f (y 1 , y 2 , y 3) = -3y 1 2 -
-3y 2 2 + 2y 3 2, where y 1 = - (2/3)x 1 + x 2 – (1/6) x 3, y 2 = (2/3)x 1 + (1/6) x 3 and y 3 = x 1 . Here there is a positive coefficient of 2 at y 3 and two negative coefficients (-3) at y 1 and y 2 (and using another method we got a positive coefficient of 2 at y 1 and two negative coefficients - (-5) at y 2 and (-1 /20) at y 3).

It should also be noted that the rank of a matrix of quadratic form, called rank of quadratic form, is equal to the number of non-zero coefficients canonical form and does not change under linear transformations.

The quadratic form f(X) is called positively (negative) certain, if for all values ​​of the variables that are not simultaneously equal to zero, it is positive, i.e. f(X) > 0 (negative, i.e.
f(X)< 0).

For example, the quadratic form f 1 (X) = x 1 2 + x 2 2 is positive definite, because is a sum of squares, and the quadratic form f 2 (X) = -x 1 2 + 2x 1 x 2 - x 2 2 is negative definite, because represents it can be represented as f 2 (X) = -(x 1 - x 2) 2.

In most practical situations, it is somewhat more difficult to establish the definite sign of a quadratic form, so for this we use one of the following theorems (we will formulate them without proof).

Theorem. A quadratic form is positive (negative) definite if and only if all eigenvalues ​​of its matrix are positive (negative).

Theorem (Sylvester criterion). A quadratic form is positive definite if and only if all the leading minors of the matrix of this form are positive.

Main (corner) minor The kth order matrix A of the nth order is called the determinant of the matrix, composed of the first k rows and columns of the matrix A ().

Note that for negative definite quadratic forms the signs of the principal minors alternate, and the first-order minor must be negative.

For example, let us examine the quadratic form f(x 1, x 2) = 2x 1 2 + 4x 1 x 2 + 3x 2 2 for sign definiteness.

= (2 - l)*
*(3 - l) – 4 = (6 - 2l - 3l + l 2) – 4 = l 2 - 5l + 2 = 0; D = 25 – 8 = 17;
. Therefore, the quadratic form is positive definite.

Method 2. Principal minor of the first order of matrix A D 1 = a 11 = 2 > 0. Principal minor of the second order D 2 = = 6 – 4 = 2 > 0. Therefore, according to Sylvester’s criterion, the quadratic form is positive definite.

We examine another quadratic form for sign definiteness, f(x 1, x 2) = -2x 1 2 + 4x 1 x 2 - 3x 2 2.

Method 1. Let's construct a matrix of quadratic form A = . Characteristic equation will look like = (-2 - l)*
*(-3 - l) – 4 = (6 + 2l + 3l + l 2) – 4 = l 2 + 5l + 2 = 0; D = 25 – 8 = 17;
. Therefore, the quadratic form is negative definite.