A quadratic form is called canonical if everything, 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 are the eigenvalues ​​of the matrix A.

2. Lagrange's method - sequential selection of complete squares. For example, if

Then a similar procedure is performed with the quadratic form and so on. If in quadratic form everything but is then after preliminary transformation the case is reduced to the considered procedure. So, if, for example, then we put

3. Jacobi's method (in the case when all major minors quadratic forms are nonzero):

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

Ax + Wu + C = 0,

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

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

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

B = 0, A ≠ 0, C ≠ 0 (Ax + C = 0) - the straight line is 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 straight line equation can be represented 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), which belongs 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 directing vector of the straight line.

We obtain the parametric equations of the straight line by equating each of the ratios (3.4) to the parameter t:

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

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

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

From equations (3.6), one can go over to canonical equations by finding z from each equation and equating the obtained values:

.

From general equations(3.2) we can go over to the canonical and in another way, if we find some point of this straight 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) are normal vectors of 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.

Any equation of the first degree 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 equation of the plane.

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 simultaneously equal to 0.

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 of the coordinate planes: x = 0, y = 0, z = 0.

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

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

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

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

A point in relation to a plane can be located in the following way: to belong or not to 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, or be parallel, or be crossed.

The parallelism of line segments is preserved in projections.

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

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

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

Ellipse. An ellipse is a locus of 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 ellipse equation

where a- semi-major axis of the ellipse, b is the semi-minor axis of the ellipse. If 2 c is the distance between the foci, then between a, b and c(if a > b) there is a relation

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 an eccentricity e < 1 (так как c < a), and its focuses lie on the major axis.

The equation of the hyperbola shown in the figure.

Parameters:
a, b - semi-axes;
- distance between foci,
- eccentricity;
- asymptotes;
- directors.
The rectangle shown in the center of the figure is the main rectangle, its diagonals are asymptotes.

Definition 10.4.Canonical view the quadratic form (10.1) is called the following form:. (10.4)

Let us show that in a basis of eigenvectors, the quadratic form (10.1) takes the canonical form. Let be

- normalized eigenvectors corresponding to the eigenvalues λ 1, λ 2, λ 3 matrices (10.3) in the orthonormal basis. Then the matrix of the transition from the old basis to the new one will be the matrix

... In the new basis, the matrix BUT takes the diagonal form (9.7) (by the property of eigenvectors). Thus, transforming the coordinates using the formulas:

,

we obtain in the new basis the canonical form of the quadratic form with coefficients equal to the eigenvalues λ 1, λ 2, λ 3:

Remark 1. From a geometrical point of view, the considered transformation of coordinates is a rotation of the coordinate system, which aligns the old coordinate axes with the new ones.

Remark 2. If any eigenvalues ​​of the matrix (10.3) coincide, to the corresponding orthonormal eigenvectors one can add a unit vector orthogonal to each of them, and thus construct a basis in which the quadratic form takes the canonical form.

Let us bring to canonical form the quadratic form

x² + 5 y² + z² + 2 xy + 6xz + 2yz.

Its matrix has the form In the example considered in Lecture 9, the eigenvalues ​​and orthonormal eigenvectors of this matrix were found:

Let's compose the matrix of transition to the basis from these vectors:

(the order of the vectors is changed so that they form the right triple). We transform the coordinates using the formulas:

.

So, the quadratic form is reduced to the canonical form with the coefficients equal to the eigenvalues ​​of the matrix of the quadratic form.

Lecture 11.

Curves of the second order. Ellipse, hyperbola and parabola, their properties and canonical equations. Reduction of the second order equation to the canonical form.

Definition 11.1.Curves of the second order on a plane are the lines of intersection of a circular cone with planes that do not pass through its vertex.

If such a plane intersects all generatrices of one cavity of the cone, then in the section it turns out ellipse, at the intersection of the generatrices of both cavities - hyperbola, and if the secant plane is parallel to some generatrix, then the section of the cone is parabola.

Comment. All curves of the second order are given by equations of the second degree in two variables.

Ellipse.

Definition 11.2.Ellipse is called the set of points of the plane for which the sum of the distances to two fixed points F 1 and F tricks, there is a constant value.

Comment. When points coincide F 1 and F 2 the ellipse turns into a circle.

We derive the equation of the ellipse, choosing the Cartesian system

y M (x, y) coordinates so that the axis Oh coincided with a straight line F 1 F 2, start

r 1 r 2 coordinates - with the middle of the segment F 1 F 2. Let the length of this

segment is equal to 2 with, then in the selected coordinate system

F 1 O F 2 x F 1 (-c, 0), F 2 (c, 0). Let the point M (x, y) lies on the ellipse, and

the sum of the distances from it to F 1 and F 2 equals 2 but.

Then r 1 + r 2 = 2a, but ,

therefore, introducing the notation b² = a²- c² and carrying out simple algebraic transformations, we obtain canonical ellipse equation: (11.1)

Definition 11.3.Eccentricity ellipse is called the value e = s / a (11.2)

Definition 11.4.Headmistress D i focus ellipse F i F i about the axis OU perpendicular to axis Oh on distance a / e from the origin.

Comment. With a different choice of the coordinate system, the ellipse can be specified not by the canonical equation (11.1), but by a second-degree equation of a different kind.

Ellipse properties:

1) The ellipse has two mutually perpendicular axes of symmetry (principal axes of the ellipse) and a center of symmetry (center of the ellipse). If an ellipse is given by a canonical equation, then its principal axes are the coordinate axes, and its center is the origin. Since the lengths of the segments formed by the intersection of the ellipse with the principal axes are equal to 2 but and 2 b (2a>2b), then the main axis passing through the foci is called the major axis of the ellipse, and the second major axis is called the minor axis.

2) The whole ellipse is contained within the rectangle

3) Eccentricity of the ellipse e< 1.

Really,

4) The directrix of the ellipse is located outside the ellipse (since the distance from the center of the ellipse to the directrix is a / e, but e<1, следовательно, a / e> a, and the whole ellipse lies in a rectangle)

5) Distance ratio r i from point of ellipse to focus F i to distance d i from this point to the directrix corresponding to the focus is equal to the eccentricity of the ellipse.

Proof.

Distances from point M (x, y) before the focuses of the ellipse can be represented as follows:

Let's compose the directrix equations:

(D 1), (D 2). Then From here r i / d i = e, as required.

Hyperbola.

Definition 11.5.Hyperbole is called the set of points of the plane for which the modulus of the difference between the distances to two fixed points F 1 and F 2 of this plane, called tricks, there is a constant value.

Let us derive the canonical hyperbola equation by analogy with the derivation of the ellipse equation, using the same notation.

|r 1 - r 2 | = 2a, whence If we denote b² = c² - a², from here you can get

- canonical hyperbole equation. (11.3)

Definition 11.6.Eccentricity hyperbole is called the magnitude e = s / a.

Definition 11.7.Headmistress D i focus hyperbola F i, is called a straight line located in one half-plane with F i about the axis OU perpendicular to axis Oh on distance a / e from the origin.

Hyperbola properties:

1) The hyperbola has two axes of symmetry (the main axes of the hyperbola) and the center of symmetry (the center of the hyperbola). Moreover, one of these axes intersects the hyperbola at two points called the vertices of the hyperbola. It is called the real axis of the hyperbola (axis Oh for the canonical choice of the coordinate system). The other axis has no points in common with the hyperbola and is called its imaginary axis (in canonical coordinates, the axis OU). On both sides of it are the right and left branches of the hyperbola. The foci of the hyperbola are located on its real axis.

2) The branches of the hyperbola have two asymptotes determined by the equations

3) Along with the hyperbola (11.3), one can consider the so-called conjugate hyperbola, defined by the canonical equation

for which the real and imaginary axes are interchanged while maintaining the same asymptotes.

4) Eccentricity of hyperbola e> 1.

5) Distance ratio r i from point of hyperbola to focus F i to distance d i from this point to the directrix corresponding to the focus is equal to the eccentricity of the hyperbola.

The proof can be carried out in the same way as for the ellipse.

Parabola.

Definition 11.8.Parabola is called the set of points of the plane for which the distance to some fixed point F this plane is equal to the distance to some fixed straight line. Dot F called focus parabolas, and the straight line is its headmistress.

To derive the parabola equation, we choose the Cartesian

coordinate system so that its origin is the middle

D M (x, y) perpendicular FD out of focus on direct

r cy, and the coordinate axes were located parallel and

perpendicular to the directrix. Let the length of the segment FD

D O F x equals R... Then from the equality r = d follows that

because the

By algebraic transformations, this equation can be reduced to the form: y² = 2 px, (11.4)

called the canonical parabola equation... The quantity R called parameter parabolas.

Parabola properties:

1) The parabola has an axis of symmetry (parabola axis). The point of intersection of the parabola with the axis is called the apex of the parabola. If a parabola is given by a canonical equation, then its axis is the axis Oh, and the vertex is the origin.

2) The whole parabola is located in the right half-plane of the plane Ohhu.

Comment. Using the properties of directrix ellipse and hyperbola and the definition of a parabola, we can prove the following statement:

The set of points in the plane for which the ratio e the distance to some fixed point to the distance to some straight line is a constant value, it is an ellipse (for e<1), гиперболу (при e> 1) or a parabola (for e=1).

Similar information.

Reduction of the quadratic form to the canonical form.

Canonical and normal forms of a quadratic form.

Linear transformations of variables.

The concept of a quadratic form.

Quadratic forms.

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

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

Example 1:

If similar terms have already been reduced in quadratic form, then the coefficients at are denoted, and at () -. 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 the quadratic forms of Example 1 are:

The quadratic matrix of example 2:

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

where the coefficients form a non-degenerate matrix.

If variables are considered as coordinates of a vector in Euclidean space with respect to some basis, then linear transformation (2) can be considered as a transition in this space to a new basis, with respect 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 the variables in the quadratic form (1) are subjected to a linear transformation (2), then the quadratic form in 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 kind of quadratic form is called canonical... In this case, a quadratic matrix is ​​diagonal:.

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

Example: Equation of the second-order central curve by 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 squares of variables, then using a linear transformation it can be reduced to a form containing the square of at least one variable.

Proof: By condition, the quadratic form contains only terms with products of variables. Let it be nonzero for any different values ​​of i and j, i.e. - one of these terms included in the quadratic form. If you perform a linear transformation, and do not change all the others, i.e. (the determinant of this transformation is nonzero), then even two terms with squares of variables will appear in quadratic form:. These terms cannot disappear when such terms are reduced, since each of the remaining terms contains at least one variable other than either from or from.

Example:

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

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

We denote

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

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

The expression on the right does not contain a variable and is a quadratic form in variables. Let's denote this expression through g, and the coefficient through, and then f will be equal to:. If we perform a linear transformation:, the determinant of which is nonzero, then g will be a quadratic form in variables, and the quadratic form f will be reduced to the form (2). The lemma is proved.

Theorem: Any quadratic form can be converted to canonical form using variable transformations.

Proof: We use induction on the number of variables. The quadratic form of is:, which is already canonical. Suppose the theorem is true for a quadratic form in n-1 variables and prove that it is true for a quadratic form in n variables.

If f does not contain squares of variables, then by Lemma 1 it can be reduced to the form containing the square of at least one variable; by Lemma 2, the resulting quadratic form can be represented in the form (2). Because the quadratic form is dependent on n-1 variables, then according to the inductive assumption it can be reduced to the canonical form using a linear transformation of these variables to variables, if we add a formula to the formulas of this transition, then we get the formula for the linear transformation, which leads to the 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 a square of some variable, then Lemma 1 does not need to be applied. The above method is called by the Lagrange method.

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

Example: Bring the quadratic form to the canonical form by the Lagrange method:

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

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 the transformations (2) and (3).

Now, with the help of these transformations, let's 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 canonical form (4) using transformation

you can go to normal view:

Linear transformation, which brings the quadratic form (1) to the normal form, is expressed by the formulas:

Bibliography:

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

2. Beklemishev DV Course of analytic geometry and linear algebra. Moscow: Fizmatlit, 2006, 304 p.

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

Lecture number 26 (II semester)

Topic: The law of inertia. Positive definite forms.