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.

The number of positive terms 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), the general equation of a quadric is

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 or complex numbers are most often considered. 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 of 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 the order of their relative positions is maintained. 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, so that vector MM 1 is equal to 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 of a given vector a by 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 a ray moves, the ray translates into itself, then this movement is either an identical 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. Identity 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, groups of self-compositions of figures play an important role. 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 an equilateral triangle, the group of plane movements that transform the triangle into itself consists of 6 elements: rotations through angles around a point and symmetries about three straight lines.

They are shown in Fig. 1 with red lines. The elements of the group of self-alignments of a 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).

The transition from a vector space of 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. The main feature of projective geometry is the principle of duality, which adds elegant symmetry to many designs. Projective geometry can be studied both from a purely geometric point of view, and from an analytical (using homogeneous coordinates) and salgebraic point of view, 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 values ​​of angles, segments, areas), and the equivalence of figures is equivalent to their congruence(i.e., when figures can be translated into one another through motion while preserving metric properties), there are more “deep-lying” properties of geometric figures that are preserved under transformations of a more general type than motion. 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 geometry by providing beautiful and simple solutions to many problems complicated by the presence of parallel lines. The projective theory of conic sections is especially simple and elegant.

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 a different set of axioms.

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 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. By supplementing the affine plane with an improper line, we ensured that mapping (I.21) became 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 an 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 we perform a central projection of the plane α onto some plane α 1, then a 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. Any projective transformation is a composition of a chain of central and parallel projections. An affine transformation is a special case of a projective transformation, in which the line at infinity 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.

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 the kinetic energy of a system 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 which it is important to find out how this function in the neighborhood of a given point deviates from the linear function that approximates it. 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 for a point to give a maximum or minimum of a function is that 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 a larger number of variables are studied in a similar way.

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 through 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, orthogonal transformations are of greatest interest, i.e. those that correspond to the transition from one system of variable 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 make a linear transformation of 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.

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 the quadratic form in matrix 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 is a non-singular matrix of 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 coefficientsa 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’s bring to canonical form the quadratic 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, first select a complete square with the 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 formf(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 canonical form in different ways 1). However, canonical forms obtained by various methods have a number of common 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 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 for y 3 and two negative coefficients (-3) for y 1 and y 2 (and using another method, we got a positive coefficient of 2 for y 1 and two negative ones - (-5) for y 2 and (-1/20) for 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 nonzero coefficients of the 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 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 in the formf 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 k-th order matrices of the An-th order are 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 -)* *(3 -) – 4 = (6 - 2- 3+ 2) – 4 = 2 - 5+ 2 = 0;D= 25 – 8 = 17; . Therefore, the quadratic form is positive definite.

Method 2. Principal minor of the first order of the matrix A  1 =a 11 = 2 > 0. Principal minor of the second order  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 = . The characteristic equation will have the form = (-2 -)* *(-3 -) – 4 = (6 + 2+ 3+ 2) – 4 = 2 + 5+ 2 = 0;D= 25 – 8 = 17 ; . Therefore, the quadratic form is negative definite.

Method 2. Principal minor of the first order of the matrix A  1 =a 11 = = -2< 0. Главный минор второго порядка 2 = = 6 – 4 = 2 >0. Therefore, according to Sylvester’s criterion, the quadratic form is negative definite (the signs of the principal minors alternate, starting with the minus).

And as another example, we examine the sign-determined quadratic form 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 = . The characteristic equation will have the form = (2 -)* *(-3 -) – 4 = (-6 - 2+ 3+ 2) – 4 = 2 +- 10 = 0;D= 1 + 40 = 41; . One of these numbers is negative and the other is positive. The signs of the eigenvalues ​​are different. Consequently, the quadratic form can be neither negatively nor positively definite, i.e. this quadratic form is not sign-definite (it can take values ​​of any sign).

Method 2. Principal minor of the first order of matrix A  1 =a 11 = 2 > 0. Principal minor of the second order 2 = = -6 – 4 = -10< 0. Следовательно, по критерию Сильвестра квадратичная форма не является знакоопределенной (знаки главных миноров разные, при этом первый из них – положителен).

1The considered method of reducing a quadratic form to canonical form is convenient to use when non-zero coefficients are encountered with the squares of variables. If they are not there, it is still possible to carry out the conversion, but you have to use some other techniques. For example, let f(x 1, x 2) = 2x 1 x 2 = x 1 2 + 2x 1 x 2 + x 2 2 - x 1 2 - x 2 2 =

= (x 1 + x 2) 2 - x 1 2 - x 2 2 = (x 1 + x 2) 2 – (x 1 2 - 2x 1 x 2 + x 2 2) - 2x 1 x 2 = (x 1 + x 2) 2 – - (x 1 - x 2) 2 - 2x 1 x 2 ; 4x 1 x 2 = (x 1 + x 2) 2 – (x 1 - x 2) 2 ;f(x 1, x 2) = 2x 1 x 2 = (1/2)* *(x 1 + x 2 ) 2 – (1/2)*(x 1 - x 2) 2 =f(y 1 ,y 2) = (1/2)y 1 2 – (1/2)y 2 2, where y 1 = x 1 + x 2, аy 2 = x 1 – x 2.

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 the quadratic form in matrix 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 is a non-singular matrix of 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, first select a complete square with the 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 canonical form in different ways). However, canonical forms obtained by various methods have a number of common 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 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 nonzero coefficients of the 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 = . The characteristic equation will have the form = (-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.

In this section we will focus on a special but important class of positive quadratic forms.

Definition 3. A real quadratic form is called non-negative (non-positive) if, for any real values ​​of the variables

. (35)

In this case, the symmetric matrix of coefficients is called positive semidefinite (negative semidefinite).

Definition 4. A 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.

Let a non-negative form be given. Let's imagine it as a sum of independent squares:

. (37)

In this representation, all squares must be positive:

. (38)

Indeed, if there were any , then it would be possible to select values ​​such that

But then, with these values ​​of the variables, the form would have a negative value, which is impossible by condition. Obviously, conversely, from (37) and (38) it follows that the form is positive.

Thus, a non-negative quadratic form is characterized by the equalities.

Let now be a positive definite form. Then it is a non-negative form. Therefore, it can be represented in the form (37), where all are positive. From the positive definiteness of the form it follows that . Indeed, in the case it is possible to select values ​​that are not simultaneously equal to zero, at which all would turn to zero. But then, by virtue of (37), at , which contradicts condition (36).

It is easy to see that conversely, if in (37) and are all positive, then it is a positive definite form.

In other words, a non-negative form is positive definite if and only if it is not singular.

The following theorem gives a criterion for the positive definiteness of a form in the form of inequalities that the form coefficients must satisfy. In this case, the notation already encountered in previous paragraphs for successive principal minors of the matrix is ​​used:

.

Theorem 3. In order for a quadratic form to be positive definite, it is necessary and sufficient that the inequalities be satisfied

Proof. The sufficiency of conditions (39) follows directly from Jacobi formula (28). The necessity of conditions (39) is established as follows. From the positive definiteness of the form follows the positive definiteness of “truncated” forms

.

But then all these forms must be non-singular, i.e.

Now we have the opportunity to use the Jacobi formula (28) (at ). Since on the right side of this formula all squares must be positive, then

This implies inequalities (39). The theorem has been proven.

Since any principal minor of a matrix, with proper renumbering of variables, can be placed in the upper left corner, then we have

Consequence. In positive definite quadratic form, all major minors of the coefficient matrix are positive:

Comment. From the non-negativity of successive principal minors

the non-negativity of the form does not follow. Indeed, the form

,

wherein , satisfies the conditions , but is not non-negative.

However, the following holds

Theorem 4. In order for a quadratic form to be non-negative, it is necessary and sufficient that all the major minors of its coefficient matrix are non-negative:

Proof. Let us introduce the auxiliary form was non-positive, it is necessary and sufficient for the inequalities to take place