Introduction…………………………………………………………….................................. .................3

1 Theoretical information about quadratic forms……………………………4

1.1 Definition of quadratic form……………………………………….…4

1.2 Reducing the quadratic form to canonical form………………...6

1.3 Law of inertia…………………………………………………………….….11

1.4 Positive definite forms……………………………………...18

2 Practical use quadratic forms …………………………22

2.1 Solving typical problems………………………………………………………………22

2.2 Tasks for independent decision……...………………….………...26

2.3 Test tasks………………………………………………………………...27

Conclusion………….……………………………...…………………………29

List of used literature……………………………………………………...30

INTRODUCTION

Initially, the theory of quadratic forms was used to study curves and surfaces defined by second-order equations containing two or three variables. Later, this theory found other applications. In particular, when mathematical modeling economic processes, the objective functions may contain quadratic terms. Numerous applications of quadratic forms have required the construction general theory, when the number of variables is equal to any

, and the coefficients of the quadratic form are not always real numbers.

The theory of quadratic forms was first developed by the French mathematician Lagrange, who owned many ideas in this theory, in particular, he introduced important concept reduced form, with the help of which he proved the finiteness of the number of classes of binary quadratic forms of a given discriminant. Then this theory was significantly expanded by Gauss, who introduced many new concepts, on the basis of which he was able to obtain proofs of difficult and deep theorems of number theory that eluded his predecessors in this field.

The purpose of the work is to study the types of quadratic forms and ways to reduce quadratic forms to canonical form.

This work sets the following tasks: select the necessary literature, consider definitions, solve a number of problems and prepare tests.

1 THEORETICAL INFORMATION ABOUT QUADRATIC FORMS

1.1 DEFINITION OF QUADRATIC FORM

Quadratic shape

of unknowns is a sum, each term of which is either the square of one of these unknowns, or the product of two different unknowns. The quadratic form comes in two forms: real and complex, depending on whether its coefficients are real or complex numbers.

Denoting the coefficient at

through , and when producing , through , the quadratic form can be represented as: .

From the coefficients

it is possible to construct a square matrix of order ; it is called a matrix of quadratic form, and its rank is called the rank of the quadratic form. If, in particular, , where , that is, the matrix is ​​non-degenerate, then the quadratic form is called non-degenerate. For any symmetric matrix of order one can be specified in a fully defined quadratic form: (1.1) - unknowns, having matrix elements with their coefficients.

Let us now denote by

a column composed of unknowns: . is a matrix with rows and one column. Transposing this matrix, we obtain the matrix: , made up of one line.

Quadratic form (1.1) with matrix

can now be written as a product:.

1.2 REDUCTION TO QUADRATIC FORM

TO THE CANONICAL VIEW

Suppose that the quadratic form

from the unknowns has already been reduced by a non-degenerate linear transformation to the canonical form , where are the new unknowns. Some of the coefficients may be zero. Let us prove that the number of nonzero coefficients is necessarily equal to the rank of the form. The matrix of this quadratic form has a diagonal form ,

and the requirement that this matrix has rank

, is equivalent to the assumption that its main diagonal contains exactly nonzero elements.

Theorem. Any quadratic form can be reduced to canonical form by some non-degenerate linear transformation. If a real quadratic form is considered, then all the coefficients of the specified linear transformation can be considered real.

Proof. This theorem is true for the case of quadratic forms in one unknown, since any such form has the form

, which is canonical. Let us introduce a proof by induction, that is, prove the theorem for quadratic forms in unknowns, considering that it has already been proven for forms with a smaller number of unknowns.

Let the quadratic form (1.1) of

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 the order curve (or surface). 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, orthogonal transformations are of greatest interest, 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.

When solving various applied problems, it is often necessary to study quadratic forms.

Definition. A quadratic form L(, 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:

L( ,x 2 ,...,x n) =

We assume that the coefficients of the quadratic form are real numbers, and

The matrix A = () (i, j = 1, 2, ..., n), composed of these coefficients, is called a matrix of quadratic form.

In matrix notation, the quadratic form has the form: L = X"AX, where X = (x 1, x 2,..., x n)" - matrix-column of variables.

Example 8.1

Write the quadratic form L( , x 2 , x 3) = in matrix form.

Let's find a matrix of quadratic form. Its diagonal elements are equal to the coefficients of the squared variables, i.e. 4, 1, -3, and other elements - to the halves of the corresponding coefficients of the quadratic form. That's why

L=( , x 2 , x 3) .

With a non-degenerate linear transformation X = CY, the matrix of quadratic form takes the form: A * = C "AC. (*)

Example 8.2

Given the quadratic form L(x x, x 2) =2x 1 2 +4x 1 x 2 -3. Find the quadratic form L(y 1 ,y 2) obtained from the given linear transformation = 2у 1 - 3y 2 , x 2 = y 1 + y 2.

The matrix of a given quadratic form is A= , and the linear transformation matrix is

C = . Therefore, according to (*) matrix of the required quadratic form

And the quadratic form looks like

L(y 1, y 2) = .

It should be noted that with some well-chosen linear transformations, the form of the quadratic form can be significantly simplified.

Definition. The quadratic form L(,x 2,...,x n) = is called canonical (or has a canonical form) if all its coefficients = 0 for i¹j:

L= , and its matrix is ​​diagonal.

The following theorem is true.

Theorem. Any quadratic form can be reduced to canonical form using a non-degenerate linear transformation of variables.

Example 8.3

Reduce the quadratic form to canonical form

L( , x 2 , x 3) =

First, we select the complete square of the variable, the coefficient of the square of which is different from zero:

Now we select the perfect square for the variable whose coefficient is different from zero:

So, a non-degenerate linear transformation

reduces this quadratic form to canonical form:

The canonical form of a quadratic form is not uniquely defined, since the same quadratic form can be reduced to the canonical form in many ways. However, the received different ways canonical forms have a number of general properties. Let us formulate one of these properties as a theorem.

Theorem (law of inertia of quadratic forms). The number of terms with positive (negative) coefficients of the quadratic form does not depend on the method of reducing the form to this form.

It should be noted that the rank of a matrix of quadratic form equal to the number nonzero coefficients of the canonical form and does not change under linear transformations.

Definition. The quadratic form L(, x 2, ..., x n) is called positive (negative) definite if, for all values ​​of the variables, at least one of which is nonzero,

L( , x 2 , ..., x n) > 0 (L( , x 2 , ..., x n)< 0).

So, For example, quadratic form is positive definite, and the form is negative definite.

Theorem. In order for the quadratic form L = X"AX to be positive (negative) definite, it is necessary and sufficient that all eigenvalues ​​of matrix A are positive (negative).

Square 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 the following form:

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 immediately anticipate a gross mistake when they lose the “minus” of a 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 values).

This form is called alternating sign. And if with linear form everything is transparent, 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 is greater than 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 angular minor is of the 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 the 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.

Is there a difference with the eigenvalue method? ;)

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 solving on your own:

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 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 the 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:

Purpose of the service. Online calculator used to find Hessian matrices and determining the type of function (convex or concave) (see example). The solution is drawn up in Word format. For a function of one variable f(x), intervals of convexity and concavity are determined.

Rules for entering functions:

A twice continuously differentiable function f(x) is convex (concave) if and only if Hessian matrix the function f(x) with respect to x is positive (negative) semidefinite for all x (see points of local extrema of a function of several variables).

Function critical points:

• if the Hessian is positive definite, then x 0 is the local minimum point of the function f(x),
• if the Hessian is negative definite, then x 0 is the local maximum point of the function f(x),
• if the Hessian is not sign-definite (takes both positive and negative values) and is non-degenerate (det G(f) ≠ 0), then x 0 is the saddle point of the function f(x).

Criteria for the definiteness of a matrix (Sylvester's theorem)

Positive certainty:

• all diagonal elements of the matrix must be positive;
• all leading main qualifiers must be positive.

For positive semidefinite matrices Sylvester criterion sounds like this: A form is positive semidefinite if and only if all major minors are non-negative. If the Hessian matrix at a point is positive semidefinite (all major minors are non-negative), then this is a minimum point (however, if the Hessian is semidefinite and one of the minors is 0, then this may be a saddle point. Additional checks are needed).

Positive semi-definiteness:

• all diagonal elements are non-negative;
• all main determinants are non-negative.

The major determinant is the determinant of the major minor.

A square symmetric matrix of order n, the elements of which are the partial derivatives of the second-order objective function, called the Hessian matrix and is designated:

In order for a symmetric matrix to be positive definite, it is necessary and sufficient that all its diagonal minors are positive, i.e.


for the matrix A = (a ij) are positive.

Negative certainty.
In order for a symmetric matrix to be negative definite, it is necessary and sufficient that the following inequalities take place:
(-1) k D k > 0, k=1,.., n.
In other words, in order for the quadratic form to be negative definite, it is necessary and sufficient that the signs of the angular minors of a matrix of quadratic form alternate, starting with the minus sign. For example, for two variables, D 1< 0, D 2 > 0.

If the Hessian is semidefinite, then this may also be an inflection point. Needed additional research, which can be carried out according to one of the following options:

1. Decreasing order. A change of variables is made. For example, for a function of two variables it is y=x, as a result we get a function of one variable x. Next, we examine the behavior of the function on the lines y=x and y=-x. If in the first case the function at the point under study will have a minimum, and in the other case a maximum (or vice versa), then the point under study is a saddle point.
2. Finding the eigenvalues ​​of the Hessian. If all values ​​are positive, the function at the point under study has a minimum, if all values ​​are negative, there is a maximum.
3. Study of the function f(x) in the neighborhood of the point ε. Variables x are replaced by x 0 +ε. Next, it is necessary to prove that the function f(x 0 +ε) of one variable ε is either greater than zero (then x 0 is the minimum point) or less than zero (then x 0 is the maximum point).

Note. To find inverse Hessian it is enough to find the inverse matrix.

Example No. 1. Which of the following functions are convex or concave: f(x) = 8x 1 2 +4x 1 x 2 +5x 2 2 .
Solution. 1. Let's find partial derivatives.


2. Let's solve the system of equations.
-4x 1 +4x 2 +2 = 0
4x 1 -6x 2 +6 = 0
We get:
a) From the first equation we express x 1 and substitute it into the second equation:
x 2 = x 2 + 1/2
-2x 2 +8 = 0
Where x 2 = 4
We substitute these values ​​x 2 into the expression for x 1. We get: x 1 = 9 / 2
The number of critical points is 1.
M 1 (9 / 2 ;4)
3. Let's find the second order partial derivatives.



4. Let us calculate the value of these second-order partial derivatives at the critical points M(x 0 ;y 0).
We calculate the values ​​for point M 1 (9 / 2 ;4)



We build the Hessian matrix:

D 1 = a 11< 0, D 2 = 8 > 0
Since the diagonal minors have different signs, nothing can be said about the convexity or concavity of the function.