Canonical quadratic form. The canonical form of the quadratic form. Reducing the quadratic form to the canonical form

Given a quadratic form (2) A(x, x) \u003d, where x = (x 1 , x 2 , …, x n). Consider a quadratic form in space R 3, that is x = (x 1 , x 2 , x 3), A(x, x) =
+
+
+
+
+
+ +
+
+
=
+
+
+ 2
+ 2
+ + 2
(we used the condition of symmetry of the shape, namely and 12 = and 21 , and 13 = and 31 , and 23 = and 32). Let us write the matrix of the quadratic form A in the basis ( e}, A(e) =
... When the basis changes, the matrix of the quadratic form changes according to the formula A(f) = C t A(e)Cwhere C - matrix of transition from the basis ( e) to the basis ( f), and C t - transposed matrix C.

Definition11.12. The form of a quadratic form with a diagonal matrix is \u200b\u200bcalled canonical.

So let A(f) =
then A"(x, x) =
+
+
where x" 1 , x" 2 , x"3 - vector coordinates x in a new basis ( f}.

Definition11.13. Let in n V such basis is chosen f = {f 1 , f 2 , …, f n ), in which the quadratic form has the form

A(x, x) =
+
+ … +
, (3)

where y 1 , y 2 , …, y n - vector coordinates x in the basis ( f). Expression (3) is called canonical view quadratic form. Coefficients  1, λ 2,…, λ n are called canonical; a basis in which the quadratic form has the canonical form is called canonical basis.

Comment... If the quadratic form A(x, x) is reduced to canonical form, then, generally speaking, not all coefficients  i are nonzero. The rank of a quadratic form is equal to the rank of its matrix in any basis.

Let the rank of the quadratic form A(x, x) is equal rwhere r ≤ n... A matrix of quadratic form in canonical form has a diagonal form. A(f) =
since its rank is r, then among the coefficients  i should be rnot equal to zero. Hence it follows that the number of nonzero canonical coefficients is equal to the rank of the quadratic form.

Comment... A linear transformation of coordinates is a transition from variables x 1 , x 2 , …, x n to variables y 1 , y 2 , …, y n , in which old variables are expressed in terms of new variables with some numerical coefficients.

x 1 \u003d α 11 y 1 + α 12 y 2 + ... + α 1 n y n ,

x 2 \u003d α 2 1 y 1 + α 2 2 y 2 + ... + α 2 n y n ,

………………………………

x 1 \u003d α n 1 y 1 + α n 2 y 2 + ... + α nn y n .

Since each transformation of the basis corresponds to a nondegenerate linear transformation of coordinates, the question of reducing the quadratic form to the canonical form can be solved by choosing the appropriate nondegenerate transformation of coordinates.

Theorem 11.2 (the main theorem on quadratic forms). Any quadratic form A(x, x) given in n-dimensional vector space V, using a nondegenerate linear transformation of coordinates can be reduced to the canonical form.

Evidence... (Lagrange's method) The idea of \u200b\u200bthis method is to successively complement the square trinomial in each variable to a complete square. We will assume that A(x, x) ≠ 0 and in the basis e = {e 1 , e 2 , …, e n ) has the form (2):

A(x, x) =
.

If a A(x, x) \u003d 0, then ( a ij) \u003d 0, that is, the form is already canonical. Formula A(x, x) can be transformed so that the coefficient a 11 ≠ 0. If a 11 \u003d 0, then the squared coefficient of the other variable is nonzero, then by renumbering the variables it is possible to achieve that a 11 ≠ 0. Renumbering of variables is a non-degenerate linear transformation. If all the coefficients of the squares of the variables are equal to zero, then the required transformations are obtained as follows. Let, for example, a 12 ≠ 0 (A(x, x) ≠ 0, therefore at least one coefficient a ij ≠ 0). Consider the transformation

x 1 = y 1 – y 2 ,

x 2 = y 1 + y 2 ,

x i = y i , at i = 3, 4, …, n.

This transformation is non-degenerate, since the determinant of its matrix is \u200b\u200bnonzero
= = 2 ≠ 0.

Then 2 a 12 x 1 x 2 = 2 a 12 (y 1 – y 2)(y 1 + y 2) = 2
– 2
, that is, in the form A(x, x) squares of two variables will appear.

A(x, x) =
+ 2
+ 2
+
. (4)

We convert the allocated amount to the form:

A(x, x) = a 11
, (5)

while the coefficients a ij change to ... Consider a non-degenerate transformation

y 1 = x 1 + + … + ,

y 2 = x 2 ,

y n = x n .

Then we get

A(x, x) =
. (6).

If the quadratic form
\u003d 0, then the question of the reduction A(x, x) to the canonical form is resolved.

If this form is not equal to zero, then we repeat the reasoning, considering the transformation of coordinates y 2 , …, y n and without changing the coordinate y 1 . Obviously, these transformations will be non-degenerate. In a finite number of steps, the quadratic form A(x, x) will be reduced to canonical form (3).

Comment1. Desired transformation of original coordinates x 1 , x 2 , …, x n can be obtained by multiplying the non-degenerate transformations found in the process of reasoning: [ x] = A[y], [y] = B[z], [z] = C[t], then [ x] = AB[z] = ABC[t], i.e [ x] = M[t], where M = ABC.

Comment 2. Let A(x, x) = A(x, x) =
+
+ …+
, where  i ≠ 0, i = 1, 2, …, r, where  1\u003e 0, λ 2\u003e 0,…, λ q > 0, λ q +1 < 0, …, λ r < 0.

Consider a non-degenerate transformation

y 1 = z 1 , y 2 = z 2 , …, y q = z q , y q +1 =
z q +1 , …, y r = z r , y r +1 = z r +1 , …, y n = z n ... As a result A(x, x) will take the form: A(x, x) = + + … + – – … – which is called normal kind of quadratic form.

Example11.1. Canonicalize a quadratic form A(x, x) = 2x 1 x 2 – 6x 2 x 3 + 2x 3 x 1 .

Decision... Insofar as a 11 \u003d 0, we use the transformation

x 1 = y 1 – y 2 ,

x 2 = y 1 + y 2 ,

x 3 = y 3 .

This transformation has a matrix A =
, i.e [ x] = A[y] we get A(x, x) = 2(y 1 – y 2)(y 1 + y 2) – 6(y 1 + y 2)y 3 + 2y 3 (y 1 – y 2) =

2– 2– 6y 1 y 3 – 6y 2 y 3 + 2y 3 y 1 – 2y 3 y 2 = 2– 2– 4y 1 y 3 – 8y 3 y 2 .

Since the coefficient at is not zero, you can select the square of one unknown, let it be y 1 . Let's select all the members containing y 1 .

A(x, x) = 2(– 2 y 1 y 3) – 2– 8y 3 y 2 = 2(– 2 y 1 y 3 + ) – 2– 2– 8y 3 y 2 = 2(y 1 – y 3) 2 – 2– 2– 8y 3 y 2 .

Let's perform a transformation whose matrix is \u200b\u200bequal to B.

z 1 = y 1 – y 3 ,  y 1 = z 1 + z 3 ,

z 2 = y 2 ,  y 2 = z 2 ,

z 3 = y 3 ;  y 3 = z 3 .

B =
, [y] = B[z].

We get A(x, x) = 2– 2–– 8z 2 z 3. Let's select the members containing z 2. We have A(x, x) = 2– 2(+ 4z 2 z 3) – 2= 2– 2(+ 4z 2 z 3 + 4) + + 8 – 2 = 2– 2(z 2 + 2z 3) 2 + 6.

Performing a transformation with a matrix C:

t 1 = z 1 ,  z 1 = t 1 ,

t 2 = z 2 + 2z 3 ,  z 2 = t 2 – 2t 3 ,

t 3 = z 3 ;  z 3 = t 3 .

C =
, [z] = C[t].

Got: A(x, x) = 2– 2+ 6 the canonical form of the quadratic form, while [ x] = A[y], [y] = B[z], [z] = C[t], from here [ x] = ABC[t];

ABC =


=
... The transformation formulas are as follows

x 1 = t 1 – t 2 + t 3 ,

x 2 = t 1 + t 2 – t 3 ,

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 - eigenvalues \u200b\u200bof the matrix A.

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

Then a similar procedure is performed with the quadratic form and so on. If in quadratic form everything 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 \u003d 0,

and the constants A, B are not equal to zero at the same time. This first-order equation is called general equation of the straight line.Depending on the values \u200b\u200bof 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 straight line is parallel to the Ox axis

B \u003d 0, A ≠ 0, C ≠ 0 (Ax + C \u003d 0) - the straight 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 different 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 \u003d 0, A 2 x + B 2 y + C 2 z + D 2 \u003d 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 \u003d x 1 + mt, y \u003d y 1 + nt, z \u003d z 1 + рt. (3.5)

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

x \u003d mz + a, y \u003d nz + b. (3.6)

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

.

One can pass from general equations (3.2) to 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 \u003d x 1, y \u003d 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 \u003d 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 simultaneously equal to 0.

Special cases of equation (3.1):

1. D \u003d 0, Ax + By + Cz \u003d 0 - the plane passes through the origin.

2. C \u003d 0, Ax + By + D \u003d 0 - the plane is parallel to the Oz axis.

3. C \u003d D \u003d 0, Ax + By \u003d 0 - the plane passes through the Oz axis.

4. B \u003d C \u003d 0, Ax + D \u003d 0 - the plane is parallel to the Oyz plane.

Equations of the coordinate planes: x \u003d 0, y \u003d 0, z \u003d 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 the plane can be located as follows: belong 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, or be parallel, or be crossed.

The parallelism of line segments is preserved in projections.

If the 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 same names, 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).

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 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.