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(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, 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 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 the 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 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 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 = . Characteristic equation will look like = (-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. Consequently, 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.

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

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

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

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

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

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

2 Practical Application quadratic forms …………………………22

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

2.2 Tasks for independent solution……...………………….………...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 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 the important concept of a 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

Quadratic L shape from n variables is a sum, each term of which is either the square of one of these variables, or the product of two different variables.

Assuming that in quadratic form L The reduction of similar terms has already been done, let us introduce the following notation for the coefficients of this form: the coefficient for is denoted by , and the coefficient in the product for is denoted by . Since , the coefficient of this product could also be denoted by , i.e. The notation we introduced assumes the validity of the equality . The term can now be written in the form

and the whole quadratic form L– in the form of the sum of all possible terms, where i And j already take on values ​​independently of each other
from 1 to n:

(6.13)

The coefficients can be used to construct a square matrix of order n; it's called matrix of quadratic form L, and its rank is rank this quadratic form. If, in particular, , i.e. matrix is ​​non-degenerate, then it is a quadratic form L called non-degenerate. Since , then the elements of matrix A, symmetrical with respect to the main diagonal, are equal to each other, i.e. matrix A – symmetrical. Conversely, for any symmetric matrix A n of order one can specify a well-defined quadratic form (6.13) of n variables that have elements of matrix A with their coefficients.

The quadratic form (6.13) can be represented in matrix form using the matrix multiplication introduced in Section 3.2. Let us denote by X a column composed of variables

X is a matrix with n rows and one column. Transposing this matrix, we obtain the matrix , made up of one line. The quadratic form (6.13) with matrix can now be written as the following product:

In fact:

and the equivalence of formulas (6.13) and (6.14) is established.

Write it down 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

. ●

Let us find out how the quadratic form changes under a non-degenerate linear transformation of variables.

Note that if matrices A and B are such that their product is defined, then the equality holds:

(6.15)

Indeed, if the product AB is defined, then the product will also be defined: the number of columns of the matrix is ​​equal to the number of rows of the matrix. Matrix element standing in its i th line and j th column, in the matrix AB is located in j th line and i th column. It is therefore equal to the sum of the products of the corresponding elements j-th row of matrix A and i th column of matrix B, i.e. equal to the sum of the products of the corresponding elements of the line j th column of the matrix and i th row of the matrix. This proves equality (6.15).


Let the matrix-column variables And are related by the linear relation X = CY, where C = ( c ij) there is some non-singular matrix n-th order. Then the quadratic form

or , Where .

The matrix will be symmetric, since in view of equality (6.15), which is obviously valid for any number of factors, and equality , which is equivalent to the symmetry of matrix A, we have:

So, with a non-degenerate linear transformation X=CY, the matrix of quadratic form takes the form

Comment. The rank of a quadratic form does not change when performing a non-degenerate linear transformation.

Example. Given a quadratic form

Find the quadratic form obtained from the given linear transformation

, .

○ Matrix of a given quadratic form , and the linear transformation matrix . Therefore, according to (6.16), the matrix of the desired quadratic form

and the quadratic form has the form . ●

With some well-chosen linear transformations, the form of the quadratic form can be significantly simplified.

Quadratic shape called canonical(or has canonical view), if all its coefficients at i≠ j:

,

and its matrix is ​​diagonal.

The following theorem is true.

Theorem 6.1. Any quadratic form can be reduced to canonical form using a non-degenerate linear transformation of variables.

Example. Reduce the quadratic form to canonical form

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

.

Now let’s select the square of the variable whose squared 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 canonical forms obtained by various methods have a number of general properties. Let us formulate one of these properties as a theorem.

Theorem 6.2.(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.

For example, the quadratic form

which in the example discussed on page 131 we brought to the form

it was possible by applying a non-degenerate linear transformation

bring to mind

.

As you can see, the number of positive and negative coefficients (two and one, respectively) has been preserved.

Note that the rank of a quadratic form is equal to the number of nonzero coefficients of the canonical form.

Quadratic shape is called positive (negative) definite if, for all values ​​of the variables, at least one of which is nonzero,

().

When solving various applied problems Often we have 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, canonical forms obtained by various methods have a number of common 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 a quadratic form is equal to the number of 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).