Introduction

quadratic form canonical form equation

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 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.

In this work, the following tasks are set: select the necessary literature, consider definitions and main theorems, solve a number of problems on this topic.

Reducing a quadratic form to canonical form

The origins of the theory of quadratic forms lie in analytical geometry, namely in the theory of second-order curves (and surfaces). It is known that the equation of a second-order central curve on a plane, after moving the origin of rectangular coordinates to the center of this curve, has the form

that in the new coordinates the equation of our curve will have a “canonical” form

in this equation, the coefficient of the product of unknowns is therefore equal to zero. Transformation of coordinates (2) can obviously be interpreted as a linear transformation of unknowns, moreover, non-degenerate, since the determinant of its coefficients is equal to one. This transformation is applied to the left side of equation (1), and therefore we can say that the left side of equation (1) is transformed into the left side of equation (3) by a non-degenerate linear transformation (2).

Numerous applications required the construction of a similar theory for the case when the number of unknowns instead of two is equal to any, and the coefficients are either real or any complex numbers.

Generalizing the expression on the left side of equation (1), we arrive at the following concept.

A quadratic form of unknowns is a sum in which each term is either the square of one of these unknowns or the product of two different unknowns. A quadratic form is called real or complex depending on whether its coefficients are real or can be any complex numbers.

Assuming that the reduction of similar terms has already been done in quadratic form, we introduce the following notation for the coefficients of this form: the coefficient for is denoted by, and the coefficient of the product for is denoted by (compare with (1)!).

Since, however, 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 entire quadratic form - in the form of a sum of all possible terms, where and independently of each other take values ​​from 1 to:

in particular, when we get the term

From the coefficients one can obviously construct a square matrix of order; it is called a matrix of a quadratic form, and its rank is called the rank of this quadratic form.

If, in particular, i.e. If the matrix is ​​non-degenerate, then the quadratic form is called non-degenerate. In view of equality (4), the elements of matrix A, symmetrical with respect to the main diagonal, are equal to each other, i.e. matrix A is symmetric. Conversely, for any symmetric matrix A of order one can specify a well-defined quadratic form (5) of the unknowns, which has elements of matrix A with its coefficients.

Quadratic form (5) can be written in another form using rectangular matrix multiplication. Let us first agree on the following notation: if a square or even rectangular matrix A is given, then the matrix obtained from matrix A by transposition will be denoted by. If matrices A and B are such that their product is defined, then the equality holds:

those. the matrix obtained by transposing the product is equal to the product of matrices obtained by transposing the factors, moreover, taken in reverse order.

In fact, if the product AB is defined, then the product will also be defined, as is easy to check: the number of columns of the matrix is ​​equal to the number of rows of the matrix. The matrix element located in its th row and th column is located in the AB matrix in the th row and th column. It is therefore equal to the sum of the products of the corresponding elements of the th row of matrix A and the th column of matrix B, i.e. is equal to the sum of the products of the corresponding elements of the th column of the matrix and the th row of the matrix. This proves equality (6).

Note that matrix A then and only then will be symmetric if it coincides with its transpose, i.e. If

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

Composed of one line.

Quadratic form (5) with matrix can now be written as the following product:

Indeed, the product will be a matrix consisting of one column:

Multiplying this matrix on the left by the matrix, we get a “matrix” consisting of one row and one column, namely the right side of equality (5).

What will happen to a quadratic form if the unknowns included in it are subjected to a linear transformation

From here by (6)

Substituting (9) and (10) into entry (7) of the form, we obtain:

Matrix B will be symmetric, since in view of equality (6), which is obviously valid for any number of factors, and an equality equivalent to the symmetry of the matrix, we have:

Thus, the following theorem is proven:

The quadratic form of the unknowns, which has a matrix, after performing a linear transformation of the unknowns with the matrix turns into a quadratic form of the new unknowns, and the matrix of this form is the product.

Let us now assume that we are performing a non-degenerate linear transformation, i.e. , and therefore and are non-singular matrices. The product is obtained in this case by multiplying the matrix by non-singular matrices and therefore, the rank of this product is equal to the rank of the matrix. Thus, the rank of the quadratic form does not change when performing a non-degenerate linear transformation.

Let us now consider, by analogy with the geometric problem indicated at the beginning of the section of reducing the equation of a second-order central curve to the canonical form (3), the question of reducing an arbitrary quadratic form by some non-degenerate linear transformation to the form of a sum of squares of unknowns, i.e. to such a form when all coefficients in the products of various unknowns are equal to zero; this special type quadratic form is called canonical. Let us first assume that the quadratic form in the unknowns has already been reduced by a non-degenerate linear transformation to the canonical form

where are the new unknowns. Some of the odds may. Of course, be zeros. Let us prove that the number of nonzero coefficients in (11) is necessarily equal to the rank of the form.

In fact, since we arrived at (11) using a non-degenerate transformation, the quadratic form on the right side of equality (11) must also be of rank.

However, the matrix of this quadratic form has a diagonal form

and requiring that this matrix have rank is equivalent to requiring that its main diagonal contains exactly zero elements.

Let us proceed to the proof of the following main theorem about quadratic forms.

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.

This theorem is true for the case of quadratic forms in one unknown, since every such form has a form that is canonical. We can, therefore, carry out the proof by induction on the number of unknowns, i.e. prove the theorem for quadratic forms in n unknowns, considering it already proven for forms with a smaller number of unknowns.

Empty given quadratic form

from n unknowns. We will try to find a non-degenerate linear transformation that would separate the square of one of the unknowns, i.e. would lead to the form of the sum of this square and some quadratic form of the remaining unknowns. This goal is easily achieved if among the coefficients in the form matrix on the main diagonal there are non-zero coefficients, i.e. if (12) includes the square of at least one of the unknowns with a difference from zero coefficients

Let, for example, . Then, as is easy to check, the expression, which is a quadratic form, contains the same terms with the unknown as our form, and therefore the difference

will be a quadratic form containing only unknowns, but not. From here

If we introduce the notation

then we get

where will now be a quadratic form about the unknowns. Expression (14) is the desired expression for the form, since it is obtained from (12) by a non-degenerate linear transformation, namely the transformation inverse to linear transformation (13), which has as its determinant and is therefore not degenerate.

If there are equalities, then we first need to perform an auxiliary linear transformation, leading to the appearance of squares of unknowns in our form. Since among the coefficients in the entry (12) of this form there must be non-zero ones - otherwise there would be nothing to prove - then let, for example, i.e. is the sum of a term and terms, each of which includes at least one of the unknowns.

Let us now perform a linear transformation

It will be non-degenerate, since it has a determinant

As a result of this transformation, the member of our form will take the form

those. in the form there will appear, with non-zero coefficients, squares of two unknowns at once, and they cannot cancel with any of the other terms, since each of these latter includes at least one of the unknowns. Now we are in the conditions of the case already considered above, those. Using another non-degenerate linear transformation we can reduce the form to the form (14).

To complete the proof, it remains to note that the quadratic form depends on less than the number of unknowns and therefore, by the induction hypothesis, is reduced to a canonical form by some non-degenerate transformation of the unknowns. This transformation, considered as a (non-degenerate, as is easy to see) transformation of all unknowns, in which it remains unchanged, leads, therefore, (14) to the canonical form. Thus, the quadratic form by two or three non-degenerate linear transformations, which can be replaced by one non-degenerate transformation - their product, is reduced to the form of a sum of squares of unknowns with some coefficients. The number of these squares is equal, as we know, to the rank of the form. If, moreover, the quadratic form is real, then the coefficients both in the canonical form of the form and in the linear transformation leading to this form will be real; in fact, both the linear transformation inverse (13) and the linear transformation (15) have real coefficients.

The proof of the main theorem is complete. The method used in this proof can be applied in specific examples to actually reduce a quadratic form to its canonical form. It is only necessary, instead of induction, which we used in the proof, to consistently isolate the squares of the unknowns using the method outlined above.

Example 1. Reduce a quadratic form to canonical form

Due to the absence of squared unknowns in this form, we first perform a non-degenerate linear transformation

with matrix

after which we get:

Now the coefficients for are different from zero, and therefore from our form we can isolate the square of one unknown. Believing

those. performing a linear transformation for which the inverse will have a matrix

we will bring to mind

So far, only the square of the unknown has been isolated, since the form still contains the product of two other unknowns. Using the inequality of the coefficient at to zero, we will once again apply the method outlined above. Performing a linear transformation

for which the inverse has the matrix

we will finally bring the form to the canonical form

A linear transformation that immediately leads (16) to the form (17) will have as its matrix the product

You can also check by direct substitution that the non-degenerate (since the determinant is equal) linear transformation

turns (16) into (17).

The theory of reducing a quadratic form to canonical form is constructed by analogy with the geometric theory of central curves of the second order, but cannot be considered a generalization of this latter theory. In fact, our theory allows the use of any non-degenerate linear transformations, while bringing a second-order curve to its canonical form is achieved by using linear transformations of a very special type,

being the rotation of the plane. This geometric theory can, however, be generalized to the case of quadratic forms in unknowns with real coefficients. An exposition of this generalization, called the reduction of quadratic forms to the principal axes, will be given below.

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.

1. Orthogonal transformation of space:

Where - eigenvalues ​​of the matrix A.

2. 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

3. Jacobi method (in the case when all major minors quadratic form are different from zero):

Any straight line on the plane can be specified by a first-order equation

Ax + Wu + C = 0,

Moreover, the constants A and B are not equal to zero at the same time. This first order equation is called general equation of a straight line. Depending on the values constant A, B and C the following special cases are possible:

C = 0, A ≠0, B ≠ 0 – the straight line passes through the origin

A = 0, B ≠0, C ≠0 (By + C = 0) - straight line parallel to the Ox axis

B = 0, A ≠0, C ≠ 0 (Ax + C = 0) – straight line parallel to the Oy axis

B = C = 0, A ≠0 – the straight line coincides with the Oy axis

A = C = 0, B ≠0 – the straight line coincides with the Ox axis

The equation of a straight line can be represented in in various forms depending on any given initial conditions.

A straight line in space can be specified:

1) as a line of intersection of two planes, i.e. system of equations:

A 1 x + B 1 y + C 1 z + D 1 = 0, A 2 x + B 2 y + C 2 z + D 2 = 0; (3.2)

2) by its two points M 1 (x 1, y 1, z 1) and M 2 (x 2, y 2, z 2), then the straight line passing through them is given by the equations:

= ; (3.3)

3) the point M 1 (x 1, y 1, z 1) belonging to it, and the vector a(m, n, p), collinear to it. Then the straight line is determined by the equations:

. (3.4)

Equations (3.4) are called canonical equations of the line.

Vector a called direction vector straight.

Parametric equations we obtain a straight line by equating each of the relations (3.4) to the parameter t:

x = x 1 +mt, y = y 1 + nt, z = z 1 + rt. (3.5)

Solving system (3.2) as a system linear equations relatively unknown x And y, we arrive at the equations of the line in projections or to given equations of the straight line:

x = mz + a, y = nz + b. (3.6)

From equations (3.6) we can go to canonical equations, finding z from each equation and equating the resulting values:

.

From general equations (3.2) you can go to canonical ones in another way, if you find any point on this line and its direction vector n= [n 1 , n 2 ], where n 1 (A 1, B 1, C 1) and n 2 (A 2 , B 2 , C 2) - normal vectors given planes. If one of the denominators m, n or r in equations (3.4) turns out to be equal to zero, then the numerator of the corresponding fraction must be set equal to zero, i.e. system

is equivalent to the system ; such a straight line is perpendicular to the Ox axis.

System is equivalent to the system x = x 1, y = y 1; the straight line is parallel to the Oz axis.

Every first degree equation with respect to coordinates x, y, z

Ax + By + Cz +D = 0 (3.1)

defines a plane, and vice versa: any plane can be represented by equation (3.1), which is called plane equation.

Vector n(A, B, C) orthogonal to the plane is called normal vector plane. In equation (3.1), the coefficients A, B, C are not equal to 0 at the same time.

Special cases of equation (3.1):

1. D = 0, Ax+By+Cz = 0 - the plane passes through the origin.

2. C = 0, Ax+By+D = 0 - the plane is parallel to the Oz axis.

3. C = D = 0, Ax + By = 0 - the plane passes through the Oz axis.

4. B = C = 0, Ax + D = 0 - the plane is parallel to the Oyz plane.

Equations coordinate planes: x = 0, y = 0, z = 0.

A straight line may or may not belong to a plane. It belongs to a plane if at least two of its points lie on the plane.

If a line does not belong to the plane, it can be parallel to it or intersect it.

A line is parallel to a plane if it is parallel to another line lying in that plane.

A straight line can intersect a plane at different angles and, in particular, be perpendicular to it.

A point in relation to the plane can be located in the following way: belong to it or not belong to it. A point belongs to a plane if it is located on a straight line located in this plane.

In space, two lines can either intersect, be parallel, or be crossed.

The parallelism of line segments is preserved in projections.

If the lines intersect, then the points of intersection of their projections of the same name are on the same connection line.

Crossing lines do not belong to the same plane, i.e. do not intersect or parallel.

in the drawing, the projections of lines of the same name, taken separately, have the characteristics of intersecting or parallel lines.

Ellipse. An ellipse is called locus points for which the sum of the distances to two fixed points (foci) is the same constant value for all points of the ellipse (this constant value must be greater than the distance between the foci).

The simplest equation of an ellipse

Where a - semi-major axis ellipse, b- semiminor axis of the ellipse. If 2 c- distance between focuses, then between a, b And c(If a > b) there is a relationship

a 2 - b 2 = c 2 .

The eccentricity of an ellipse is the ratio of the distance between the foci of this ellipse to the length of its major axis

The ellipse has eccentricity e < 1 (так как c < a), and its foci lie on the major axis.

Equation of the hyperbola shown in the figure.

Parameters:
a, b – semi-axes;
- distance between focuses,
- eccentricity;
- asymptotes;
- headmistresses.
The rectangle shown in the center of the picture is the main rectangle; its diagonals are asymptotes.

When considering Euclidean space, we introduced the definition of a quadratic form. Using some matrix

a second-order polynomial of the form is constructed

which is called the quadratic form generated by a square matrix A.

Quadratic forms are closely related to second-order surfaces in n-dimensional Euclidean space. The general equation of such surfaces in our three-dimensional Euclidean space in the Cartesian coordinate system has the form:

The top line is nothing more than the quadratic form, if we put x 1 =x, x 2 =y, x 3 =z:

- symmetric matrix (a ij = a ji)

Let us assume for generality that the polynomial

there is a linear form. Then general equation surface is the sum of a quadratic form, a linear form and some constant.

The main task of the theory of quadratic forms is to reduce the quadratic form to the maximum simple view using a non-degenerate linear transformation of variables or, in other words, a change of basis.

Let us remember that when studying second-order surfaces, we came to the conclusion that by rotating the coordinate axes we can get rid of terms containing the product xy, xz, yz or x i x j (ij). Further, by parallel translation of the coordinate axes, you can get rid of the linear terms and ultimately reduce the general surface equation to the form:

In the case of a quadratic form, reducing it to the form

is called reducing a quadratic form to canonical form.

Rotation of coordinate axes is nothing more than replacing one basis with another, or, in other words, a linear transformation.

Let's write the quadratic form in matrix form. To do this, let's imagine it as follows:

L(x,y,z) = x(a 11 x+a 12 y+a 13 z)+

Y(a 12 x+a 22 y+a 23 z)+

Z(a 13 x+a 23 y+a 33 z)

Let's introduce a matrix - column

Then
- whereX T =(x,y,z)

Matrix notation of quadratic form. This formula is obviously valid in the general case:

The canonical form of the quadratic form obviously means that the matrix A has a diagonal appearance:

Consider some linear transformation X = SY, where S - square matrix order n, and the matrices - columns X and Y are:

The matrix S is called the linear transformation matrix. Let us note in passing that any matrix of nth order with a given basis corresponds to a certain linear operator.

The linear transformation X = SY replaces the variables x 1, x 2, x 3 with new variables y 1, y 2, y 3. Then:

where B = S T A S

The task of reduction to canonical form comes down to finding a transition matrix S such that matrix B takes on a diagonal form:

So, quadratic form with matrix A after linear transformation of variables goes into quadratic form from new variables with matrix IN.

Let's turn to linear operators. Each matrix A for a given basis corresponds to a certain linear operator A . This operator obviously has a certain system of eigenvalues ​​and eigenvectors. Moreover, we note that in Euclidean space the system of eigenvectors will be orthogonal. We proved in the previous lecture that in the eigenvector basis the matrix of a linear operator has a diagonal form. Formula (*), as we remember, is the formula for transforming the matrix of a linear operator when changing the basis. Let us assume that the eigenvectors of the linear operator A with matrix A - these are the vectors y 1, y 2, ..., y n.

And this means that if the eigenvectors y 1, y 2, ..., y n are taken as a basis, then the matrix of the linear operator in this basis will be diagonal

or B = S -1 A S, where S is the transition matrix from the initial basis ( e) to basis ( y). Moreover, in an orthonormal basis, the matrix S will be orthogonal.

That. to reduce the quadratic form to the canonical form, it is necessary to find the eigenvalues ​​and eigenvectors of the linear operator A, which has in the initial basis the matrix A, which generates the quadratic form, go to the basis of the eigenvectors and in new system coordinates, construct a quadratic form.

Let's look at specific examples. Let's consider second order lines.

or

By rotating the coordinate axes and subsequent parallel translation of the axes, this equation can be reduced to the form (variables and coefficients are redesignated x 1 = x, x 2 = y):

1)
if the line is central, 1  0,  2  0

2)
if the line is non-central, i.e. one of i = 0.

Let us recall the types of second-order lines. Center lines:

Off-center lines:

5) x 2 = a 2 two parallel lines;

6) x 2 = 0 two merging lines;

7) y 2 = 2px parabola.

Cases 1), 2), 7) are of interest to us.

Let's look at a specific example.

Bring the equation of the line to canonical form and construct it:

5x 2 + 4xy + 8y 2 - 32x - 56y + 80 = 0.

The matrix of quadratic form is
. Characteristic equation:

Its roots:

Let's find the eigenvectors:

When  1 = 4:
u 1 = -2u 2 ; u 1 = 2c, u 2 = -c or g 1 = c 1 (2 i – j).

When  2 = 9:
2u 1 = u 2 ; u 1 = c, u 2 = 2c or g 2 = c 2 ( i+2j).

We normalize these vectors:

Let's create a linear transformation matrix or a transition matrix to the basis g 1, g 2:

- orthogonal matrix!

The coordinate transformation formulas have the form:

or

Let's substitute lines into our equation and get:

Let's make a parallel translation of the coordinate axes. To do this, select complete squares of x 1 and y 1:

Let's denote
. Then the equation will take the form: 4x 2 2 + 9y 2 2 = 36 or

This is an ellipse with semi-axes 3 and 2. Let's determine the angle of rotation of the coordinate axes and their shift in order to construct an ellipse in the old system.

P sharp:

Check: at x = 0: 8y 2 - 56y + 80 = 0 y 2 – 7y + 10 = 0. Hence y 1,2 = 5; 2

When y = 0: 5x 2 – 32x + 80 = 0 There are no roots here, i.e. there are no points of intersection with the axis X!

Reducing a quadratic form to canonical form.

Canonical and normal form of quadratic form.

Linear transformations of variables.

The concept of quadratic form.

Square shapes.

Definition: The quadratic form of variables is a homogeneous polynomial of the second degree with respect to these variables.

Variables can be thought of as affine coordinates points of arithmetic space A n or as coordinates of a vector of n-dimensional space V n . We will denote the quadratic form of variables as.

Example 1:

If similar terms have already been reduced in quadratic form, then the coefficients for are denoted, and for () - . Thus, it is believed that. The quadratic form can be written as follows:

Example 2:

System matrix (1):

- called matrix of quadratic form.

Example: The matrices of quadratic forms of Example 1 have the form:

Example 2 quadratic form matrix:

Linear transformation of variables call such a transition from a system of variables to a system of variables in which old variables are expressed through new ones using the forms:

where the coefficients form a non-singular matrix.

If variables are considered as the coordinates of a vector in Euclidean space relative to some basis, then linear transformation (2) can be considered as a transition in this space to a new basis, relative to which the same vector has coordinates.

In what follows, we will consider quadratic forms only with real coefficients. We will assume that the variables take only real values. If in quadratic form (1) the variables are subjected to a linear transformation (2), then a quadratic form of the new variables will be obtained. In what follows, we will show that with an appropriate choice of transformation (2), the quadratic form (1) can be reduced to a form containing only the squares of the new variables, i.e. . This type of quadratic form is called canonical. The matrix of quadratic form in this case is diagonal: .

If all coefficients can take only one of the values: -1,0,1 the corresponding type is called normal.

Example: Equation of the central curve of the second order using the transition to a new coordinate system

can be reduced to the form: , and the quadratic form in this case will take the form:

Lemma 1: If the quadratic form(1)does not contain the squares of the variables, then using a linear transformation it can be brought into a form containing the square of at least one variable.

Proof: By convention, the quadratic form contains only terms with products of variables. Let for any different values ​​of i and j be different from zero, i.e. is one of these terms included in the quadratic form. If you perform a linear transformation and leave everything else unchanged, i.e. (the determinant of this transformation is different from zero), then even two terms with squares of variables will appear in quadratic form: . These terms cannot disappear when similar terms are added, because each of the remaining terms contains at least one variable different from or from.

Example:

Lemma 2: If square shape (1) contains a term with the square of the variable, for example, and at least one more term with a variable , then using a linear transformation, f can be converted to variable form , having the form: (2), Where g – quadratic form containing no variable .

Proof: Let us select in quadratic form (1) the sum of terms containing: (3) here g 1 denotes the sum of all terms that do not contain.

Let's denote

(4), where denotes the sum of all terms that do not contain.

Let us divide both sides of (4) by and subtract the resulting equality from (3), after bringing similar ones we will have:

The expression on the right side does not contain a variable and is a quadratic form of variables. Let us denote this expression by g, and the coefficient by, and then f will be equal to: . If we make a linear transformation: , whose determinant is different from zero, then g will be a quadratic form of the variables, and the quadratic form f will be reduced to the form (2). The lemma is proven.

Theorem: Any quadratic form can be reduced to canonical form using a transformation of variables.

Proof: Let us carry out induction on the number of variables. The quadratic form of has the form: , which is already canonical. Let us assume that the theorem is true for the quadratic form in n-1 variables and prove that it is true for the quadratic form in n variables.

If f does not contain squares of variables, then by Lemma 1 it can be reduced to a form containing the square of at least one variable; by Lemma 2 the resulting quadratic form can be represented in the form (2). Because quadratic form is dependent on n-1 variables, then by inductive assumption it can be reduced to canonical form using a linear transformation of these variables to variables, if we add a formula to the formulas of this transition, then we obtain formulas for a linear transformation that leads to canonical form the quadratic form contained in equality (2). The composition of all the transformations of variables under consideration is the desired linear transformation, leading to the canonical form of the quadratic form (1).

If the quadratic form (1) contains the square of any variable, then Lemma 1 does not need to be applied. The given method is called Lagrange method.

From the canonical view, where, you can go to normal looking, where, if, and, if, using the transformation:

Example: Reduce the quadratic form to canonical form using the Lagrange method:

Because Since the quadratic form f already contains the squares of some variables, Lemma 1 does not need to be applied.

We select members containing:

3. To obtain a linear transformation that directly reduces the form f to the form (4), we first find the transformations inverse to transformations (2) and (3).

Now, using these transformations, we will build their composition:

If we substitute the obtained values ​​(5) into (1), we immediately obtain a representation of the quadratic form in the form (4).

From the canonical form (4) using the transformation

you can go to the normal view:

A linear transformation that brings the quadratic form (1) to normal form is expressed by the formulas:

Bibliography:

1. Voevodin V.V. Linear algebra. St. Petersburg: Lan, 2008, 416 p.

2. Beklemishev D.V. Course of analytical geometry and linear algebra. M.: Fizmatlit, 2006, 304 p.

3. Kostrikin A.I. Introduction to algebra. part II. Fundamentals of algebra: textbook for universities, -M. : Physics and mathematics literature, 2000, 368 p.

Lecture No. 26 (II semester)

Subject: Law of inertia. Positive definite forms.

defines a curve on the plane. A group of terms is called a quadratic form, – linear form. If a quadratic form contains only squares of variables, then this form is called canonical, and the vectors of an orthonormal basis in which the quadratic form has a canonical form are called the principal axes of the quadratic form.
Matrix is called a matrix of quadratic form. Here a 1 2 =a 2 1. To reduce matrix B to diagonal form, it is necessary to take the eigenvectors of this matrix as a basis, then , where λ 1 and λ 2 are the eigenvalues ​​of matrix B.
In the basis of the eigenvectors of the matrix B, the quadratic form will have the canonical form: λ 1 x 2 1 +λ 2 y 2 1 .
This operation corresponds to the rotation of the coordinate axes. Then the origin of coordinates is shifted, thereby getting rid of the linear shape.
The canonical form of the second-order curve: λ 1 x 2 2 +λ 2 y 2 2 =a, and:
a) if λ 1 >0; λ 2 >0 is an ellipse, in particular, when λ 1 =λ 2 it is a circle;
b) if λ 1 >0, λ 2<0 (λ 1 <0, λ 2 >0) we have a hyperbole;
c) if λ 1 =0 or λ 2 =0, then the curve is a parabola and after rotating the coordinate axes it has the form λ 1 x 2 1 =ax 1 +by 1 +c (here λ 2 =0). Complementing to a complete square, we have: λ 1 x 2 2 =b 1 y 2.

Example. The equation of the curve 3x 2 +10xy+3y 2 -2x-14y-13=0 is given in the coordinate system (0,i,j), where i =(1,0) and j =(0,1).
1. Determine the type of curve.
2. Bring the equation to canonical form and construct a curve in the original coordinate system.
3. Find the corresponding coordinate transformations.

Solution. We bring the quadratic form B=3x 2 +10xy+3y 2 to the main axes, that is, to the canonical form. The matrix of this quadratic form is . We find the eigenvalues ​​and eigenvectors of this matrix:

Characteristic equation:
; λ 1 =-2, λ 2 =8. Type of quadratic form: .
The original equation defines a hyperbola.
Note that the form of the quadratic form is ambiguous. You can write 8x 1 2 -2y 1 2 , but the type of curve remains the same - a hyperbola.
We find the principal axes of the quadratic form, that is, the eigenvectors of the matrix B. .
Eigenvector corresponding to the number λ=-2 at x 1 =1: x 1 =(1,-1).
As a unit eigenvector we take the vector , where is the length of the vector x 1.
The coordinates of the second eigenvector corresponding to the second eigenvalue λ=8 are found from the system
.
1 ,j 1).
According to formulas (5) of paragraph 4.3.3. Let's move on to a new basis:
or

; . (*)

We enter the expressions x and y into the original equation and, after transformations, we get: .
Selecting complete squares: .
We carry out a parallel translation of the coordinate axes to a new origin: , .
If we introduce these relations into (*) and resolve these equalities with respect to x 2 and y 2, we obtain: , . In the coordinate system (0*, i 1, j 1) this equation has the form: .
To construct a curve, we construct a new one in the old coordinate system: the x 2 =0 axis is specified in the old coordinate system by the equation x-y-3=0, and the y 2 =0 axis by the equation x+y-1=0. The origin of the new coordinate system 0 * (2,-1) is the intersection point of these lines.
To simplify perception, we will divide the process of constructing a graph into 2 stages:
1. Transition to a coordinate system with axes x 2 =0, y 2 =0, specified in the old coordinate system by the equations x-y-3=0 and x+y-1=0, respectively.

2. Construction of a graph of the function in the resulting coordinate system.

The final version of the graph looks like this (see. Solution:Download solution

Exercise. Establish that each of the following equations defines an ellipse, and find the coordinates of its center C, semi-axis, eccentricity, directrix equations. Draw an ellipse on the drawing, indicating the axes of symmetry, foci and directrixes.
Solution.