Reduction to the canonical form of a bilinear form online. Reduction of a curve of the second order to the canonical form

Reduction of a quadratic form to a canonical form.

Canonical and normal form of a quadratic form.

Linear transformations of variables.

The concept of a quadratic form.

Square shapes.

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

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

Example 1:

If the reduction of similar terms has already been performed in the 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):

- is called quadratic matrix.

Example: Matrices of quadratic forms of example 1 have the form:

Example 2 Quadratic Matrix:

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

where the coefficients form a nonsingular matrix.

If the variables are considered as the coordinates of a vector in Euclidean space with respect to some basis, then the 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 assume that the variables take only real values. If in the 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 kind of quadratic form is called canonical. The quadratic matrix in this case 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 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 squares of variables, then by means of a linear transformation it can be reduced to a form containing the square of at least one variable.

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

Example:

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

Proof: We 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.

Denote

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

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

The expression on the right side does not contain a variable and is a quadratic form in variables. Let's denote this expression by g, and the coefficient by, and then f will be equal to: . If we perform a linear transformation: , whose determinant is different from zero, then g will be a quadratic form in 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 a 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. Assume that 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 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 the 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 get the formulas of a linear transformation that leads to the canonical form the quadratic form contained in equality (2). The composition of all 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 some 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 the normal view, where, if, and if, using the transformation:

Example: Reduce 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 transformations (2) and (3).

Now, with the help of these transformations, we will construct 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 switch back to normal:

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

Bibliography:

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

2. D. V. Beklemishev, 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 No. 26 (II semester)

Topic: Law of inertia. positive definite forms.

A quadratic form is called canonical if all i.e.

Any quadratic form can be reduced to a canonical form using linear transformations. In practice, the following methods are usually used.

1. Orthogonal transformation of space:

where - matrix eigenvalues A.

2. Lagrange's method - successive selection of full squares. For example, if

Then a similar procedure is done with the quadratic form etc. If in quadratic form everything but is then, after a preliminary transformation, the matter is reduced to the procedure considered. Thus, if, for example, then we set

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

Any line in the plane can be given by a first order equation

Ah + 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 a straight line. Depending on the values ​​of the constants A, B and C, the following special cases are possible:

C \u003d 0, A ≠ 0, B ≠ 0 - the line passes through the origin

A \u003d 0, B ≠ 0, C ≠ 0 (By + C \u003d 0) - the line is parallel to the Ox axis

B \u003d 0, A ≠ 0, C ≠ 0 ( Ax + C \u003d 0) - the line is parallel to the Oy axis

B \u003d C \u003d 0, A ≠ 0 - the straight line coincides with the Oy axis

A \u003d C \u003d 0, B ≠ 0 - the straight line coincides with the Ox axis

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

A straight line in space can be given:

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) 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), s collinear. Then the straight line is determined by the equations:

. (3.4)

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

Vector a called guide vector straight.

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

x \u003d x 1 + mt, y \u003d y 1 + nt, z \u003d z 1 + pt. (3.5)

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

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

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

.

One can pass from general equations (3.2) to canonical equations in another way, if one finds any point of 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 of the 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 tantamount to a system ; such a line is perpendicular to the x-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 plane equation.

Vector n(A, B, C) orthogonal to the plane is called normal vector planes. 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.

Coordinate plane equations: 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 may be parallel to it or intersect it.

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

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

A point in relation to a plane can be located as follows: to belong or not to belong to it. A point belongs to a plane if it is located on a line in that 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 lines intersect, then the points of intersection of their projections of the same name are on the same line of communication.

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

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

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

The simplest equation of an ellipse

where a- the major axis of the ellipse, b is the minor semiaxis of the ellipse. If 2 c- 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 foci lie on the major axis.

The equation of the hyperbola shown in the figure.

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