The quadratic form of the matrix. Positive definite quadratic forms

The concept of a quadratic form. Matrix of quadratic form. Canonical form of a quadratic form. Lagrange method. The normal form of a quadratic form. Rank, index and signature of a quadratic form. Positive definite quadratic form. Quadrics.

The concept of a quadratic form: a function on a vector space given by a homogeneous polynomial of the second degree in the coordinates of the vector.

quadratic form from n unknown is called the sum, each term of which is either the square of one of these unknowns, or the product of two different unknowns.

Quadratic Matrix: The matrix is called the matrix of quadratic form in the given basis. If the field characteristic is not equal to 2, we can assume that the matrix of the quadratic form is symmetric, that is, .

Write a matrix of quadratic form:

Consequently,

In vector-matrix form, the quadratic form is:

A , where

Canonical form of a quadratic form: 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.

Lagrange 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

The normal form of a quadratic form is: A normal quadratic form is a canonical quadratic form in which all coefficients are equal to +1 or -1.

Rank, index and signature of a quadratic form: The rank of the quadratic form BUT called the rank of the matrix BUT. The rank of a quadratic form does not change under nondegenerate transformations of the unknowns.

The number of negative coefficients is called the negative shape index.

The number of positive terms in the canonical form is called the positive index of inertia of the quadratic form, the number of negative terms is called the negative index. The difference between positive and negative indices is called the signature of the quadratic form

Positive definite quadratic form: Real quadratic form is called positive-definite (negative-definite) if for any real values of the variables that are not simultaneously equal to zero

. (36)

In this case, the matrix is also called positive definite (negative definite).

The class of positive-definite (negative-definite) forms is part of the class of non-negative (respectively, non-positive) forms.

Quads: Quadric - n-dimensional hypersurface in n+1-dimensional space, defined as the set of zeros of a polynomial of the second degree. If you enter the coordinates ( x 1 , x 2 , x n+1 ) (in Euclidean or affine space), general equation quadrics has the form

This equation can be rewritten more compactly in matrix notation:

where x = ( x 1 , x 2 , x n+1 ) is a row vector, x T is the transposed vector, Q is the size matrix ( n+1)×( n+1) (it is assumed that at least one of its elements is nonzero), P is a row vector, and R is a constant. Most often, quadrics are considered over real or complex numbers. The definition can be extended to quadrics in projective space, see below.

More generally, the set of zeros of a system of polynomial equations is known as an algebraic variety. Thus a quadric is an (affine or projective) algebraic variety of second degree and codimension 1.

Plane and space transformations.

Plane transformation definition. Definition of movement. motion properties. Two types of movements: movement of the first kind and movement of the second kind. Movement examples. Analytical expression of motion. Classification of plane motions (depending on the presence of fixed points and invariant lines). Group of plane motions.

Plane transformation definition: Definition. A plane transformation that preserves the distance between points is called movement(or displacement) of the plane. The plane transformation is called affine, if it takes any three points lying on the same line to three points also lying on the same line and at the same time preserves the simple relation of the three points.

Movement definition: This is a shape transformation that preserves the distances between points. If two figures are exactly combined with each other by means of movement, then these figures are the same, equal.

Movement properties: every orientation-preserving motion of a plane is either a parallel translation or a rotation; every orientation-changing motion of a plane is either an axial symmetry or a sliding symmetry. Points lying on a straight line, when moving, pass into points lying on a straight line, and the order of their mutual arrangement is preserved. When moving, the angles between the half-lines are preserved.

Two types of movements: movement of the first kind and movement of the second kind: Movements of the first kind are those movements that preserve the orientation of the bases of a certain figure. They can be realized with continuous movements.

Movements of the second kind are those movements that change the orientation of the bases to the opposite. They cannot be realized by continuous movements.

Examples of movements of the first kind are translation and rotation around a straight line, and movements of the second kind are central and mirror symmetries.

The composition of any number of motions of the first kind is a motion of the first kind.

The composition of an even number of movements of the second kind is a movement of the 1st kind, and the composition of an odd number of movements of the 2nd kind is a movement of the 2nd kind.

Movement examples:Parallel transfer. Let a be a given vector. Parallel transfer to the vector a is the mapping of the plane onto itself, in which each point M is mapped to the point M 1, that the vector MM 1 is equal to the vector a.

Parallel translation is a movement because it is a mapping of the plane onto itself, preserving distances. Visually, this movement can be represented as a shift of the entire plane in the direction of a given vector a by its length.

Turn . Let us designate a point O on the plane ( turning center) and set the angle α ( angle of rotation). The rotation of the plane around the point O by the angle α is the mapping of the plane onto itself, in which each point M is mapped to the point M 1, that OM = OM 1 and the angle MOM 1 is equal to α. In this case, the point O remains in its place, i.e., it is displayed in itself, and all other points rotate around the point O in the same direction - clockwise or counterclockwise (the figure shows a counterclockwise rotation).

A turn is a movement because it is a mapping of the plane onto itself, which preserves distances.

Analytical expression of movement: the analytical connection between the coordinates of the pre-image and the image of the point has the form (1).

Classification of plane motions (depending on the presence of fixed points and invariant lines): Definition:

A point in a plane is invariant (fixed) if, under a given transformation, it transforms into itself.

Example: When central symmetry the point of the center of symmetry is invariant. When turning, the point of the center of rotation is invariant. With axial symmetry, the line is invariant - the axis of symmetry is the line of invariant points.

Theorem: If the motion has no invariant point, then it has at least one invariant direction.

Example: Parallel transfer. Indeed, lines parallel to this direction are invariant as a figure as a whole, although it does not consist of invariant points.

Theorem: If some ray moves, the ray translates into itself, then this motion is either an identical transformation, or a symmetry with respect to the line containing the given ray.

Therefore, according to the presence of invariant points or figures, it is possible to classify movements.

Movement name	Invariant points	Invariant lines
Movement of the first kind.
1. - turn	(center) - 0	No
2. Identity transformation	all points of the plane	all straight
3. Central symmetry	point 0 - center	all lines passing through point 0
4. Parallel transfer	No	all straight
Movement of the second kind.
5. Axial symmetry.	set of points	axis of symmetry (straight) all straight

Plane movement group: In geometry, self-coincidence groups of figures play an important role. If - some figure on the plane (or in space), then we can consider the set of all those movements of the plane (or space), in which the figure passes into itself.

This set is a group. For example, for an equilateral triangle, the group of plane motions that take the triangle into itself consists of 6 elements: rotations by angles around a point and symmetries about three lines.

They are shown in fig. 1 with red lines. The elements of the self-coincidence group of a regular triangle can be specified in another way. To clarify this, let's number the vertices of a regular triangle with numbers 1, 2, 3. can be conditionally entered in the form of one of these brackets:

etc.

where the numbers 1, 2, 3 denote the numbers of those vertices into which vertices 1, 2, 3 pass as a result of the considered movement.

Projective spaces and their models.

Concept of projective space and model of projective space. Basic facts of projective geometry. A bunch of lines centered at point O is a projective plane model. projective points. The extended plane is a model of the projective plane. An extended three-dimensional affine or Euclidean space is a projective space model. Images of plane and spatial figures in parallel design.

Concept of projective space and model of projective space:

A projective space over a field is a space consisting of lines (one-dimensional subspaces) of some linear space over a given field. The straight spaces are called dots projective space. This definition lends itself to generalization to an arbitrary body

If it has dimension , then the dimension of the projective space is called the number , and the projective space itself is denoted and is called associated with (to indicate this, the notation is adopted).

The transition from a vector space of dimension to the corresponding projective space is called projectivization spaces.

Points can be described using homogeneous coordinates.

Basic facts of projective geometry: Projective geometry is a branch of geometry that studies projective planes and spaces. main feature projective geometry is based on the principle of duality, which adds a graceful symmetry to many designs. Projective geometry can be studied both from a purely geometric point of view, and from an analytic (using homogeneous coordinates) and salgebraic point of view, considering the projective plane as a structure over a field. Often, and historically, the real projective plane is treated as the Euclidean plane with the addition of a "line at infinity".

Whereas the properties of the figures that Euclidean geometry deals with are metric(specific values of angles, segments, areas), and the equivalence of figures is equivalent to their congruence(i.e. when figures can be translated into one another by means of movement while maintaining metric properties), there are more "deeper-lying" properties geometric shapes, which are preserved under transformations of more than general type than movement. Projective geometry studies the properties of figures that are invariant under the class projective transformations, as well as these transformations themselves.

Projective geometry complements Euclidean by providing beautiful and simple solutions for many problems complicated by the presence of parallel lines. The projective theory of conic sections is especially simple and elegant.

There are three main approaches to projective geometry: independent axiomatization, addition to Euclidean geometry, and structure over a field.

Axiomatization

A projective space can be defined using a different set of axioms.

Coxeter provides the following:

1. There is a line and a point is not on it.

2. There are at least three points on every line.

3. Exactly one straight line can be drawn through two points.

4. If A, B, C, and D different points and AB and CD intersect, then AC and BD intersect.

5. If ABC is a plane, then there is at least one point not in the plane ABC.

6. Two various planes intersect at least two points.

7. Three diagonal points of a complete quadrilateral are not collinear.

8. If there are three points on a straight line X X

The projective plane (without the third dimension) is defined by somewhat different axioms:

1. Exactly one straight line can be drawn through two points.

2. Any two lines intersect.

3. There are four points, of which there are no three collinear.

4. Three diagonal points of complete quadrilaterals are not collinear.

5. If there are three points on a straight line X are invariant under the projectivity of φ, then all points on X are invariant with respect to φ.

6. Desargues' theorem: If two triangles are perspective through a point, then they are perspective through a line.

In the presence of a third dimension, Desargues' theorem can be proved without introducing the ideal point and line.

Extended plane - projective plane model: in an affine space A3, take a bundle of lines S(O) centered at a point O and a plane Π not passing through the center of the bundle: O 6∈ Π. A bundle of lines in an affine space is a model of the projective plane. Let's set the mapping of the set of points of the plane Π to the set of lines of the bundle S (Damn, pray if you got this question, I'm sorry)

Extended three-dimensional affine or Euclidean space - projective space model:

In order to make the mapping surjective, we repeat the process of formally extending the affine plane Π to the projective plane, Π, complementing the plane Π with a set of improper points (M∞) such that: ((M∞)) = P0(O). Since in the mapping the inverse image of each plane of the bundle of planes S(O) is a line on the plane d, it is obvious that the set of all improper points of the extended plane: Π = Π ∩ (M∞), (M∞), is an improper line d∞ of the extended plane which is the inverse image of the singular plane Π0: (d∞) = P0(O) (= Π0). (I.23) Let us agree that here and below we will understand the last equality P0(O) = Π0 in the sense of equality of sets of points, but endowed with different structures. Complementing the affine plane with an improper line, we have ensured that the mapping (I.21) becomes bijective on the set of all points of the extended plane:

Images of flat and spatial figures in parallel design:

In stereometry, spatial figures are studied, but in the drawing they are depicted as flat figures. How, then, should a spatial figure be depicted on a plane? Usually in geometry, parallel design is used for this. Let p be some plane, l- a straight line intersecting it (Fig. 1). Through an arbitrary point A, not belonging to the line l draw a line parallel to the line l. The point of intersection of this line with the plane p is called the parallel projection of the point A to the plane p in the direction of the straight line l. Let's denote it A". If the point A belongs to the line l, then the parallel projection A to the plane p is considered the point of intersection of the line l with plane p.

Thus, every point A space is mapped to its projection A" onto the plane p. This correspondence is called the parallel projection onto the plane p in the direction of the straight line l.

Group of projective transformations. Application to problem solving.

The concept of projective transformation of the plane. Examples of projective plane transformations. Properties of projective transformations. Homology, properties of homology. Group of projective transformations.

The concept of a projective plane transformation: The notion of a projective transformation generalizes the notion of a central projection. If we perform the central projection of the plane α onto some plane α 1 , then the projection of α 1 onto α 2 , α 2 onto α 3 , ... and, finally, some plane α n again on α 1 , then the composition of all these projections is the projective transformation of the plane α; such a chain can include parallel projections.

Examples of projective plane transformations: A projective transformation of an augmented plane is its one-to-one mapping onto itself, which preserves the collinearity of points, or, in other words, the image of any straight line is a straight line. Any projective transformation is a composition of a chain of central and parallel projections. The affine transformation is special case projective, in which the line at infinity passes into itself.

Properties of projective transformations:

Under a projective transformation, three points not on a line are mapped to three points not on a line.

Under a projective transformation, the frame goes over to the frame.

Under a projective transformation, a line goes into a straight line, a sheaf goes into a sheaf.

Homology, homology properties:

A projective transformation of a plane that has a line of invariant points and hence a pencil of invariant lines is called a homology.

1. A line passing through corresponding noncoinciding homology points is an invariant line;

2. The lines passing through the corresponding noncoinciding homology points belong to the same pencil, the center of which is an invariant point.

3. A point, its image, and the center of homology lie on the same straight line.

Group of projective transformations: consider a projective mapping of the projective plane P 2 onto itself, that is, a projective transformation of this plane (P 2 ’ = P 2).

As before, the composition f of projective transformations f 1 and f 2 of the projective plane P 2 is the result of successive execution of transformations f 1 and f 2: f = f 2 °f 1 .

Theorem 1: The set H of all projective transformations of the projective plane P 2 is a group under the composition of projective transformations.

Quadratic forms

quadratic form f(x 1, x 2,..., x n) of n variables is called the 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 the quadratic form matrix. It's always symmetrical matrix (i.e., a matrix symmetric about the main diagonal, a ij = a ji).

In matrix notation, the quadratic form has the form 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 a quadratic form. Its diagonal elements are equal to the coefficients at the squares of the variables, and the remaining elements are equal to half 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-degenerate matrix of order n. Then the quadratic form
f(X) \u003d X T AX \u003d (CY) T A (CY) \u003d (Y T C T) A (CY) \u003d Y T (C T AC) Y.

Thus, under a non-degenerate linear transformation C, the matrix of the 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 a linear transformation.

The quadratic form is called canonical(It has canonical view) if all its coefficients a 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(the proof is not given here). Any quadratic form can be reduced to a canonical form using a non-degenerate linear transformation.

For example, let us reduce to the canonical form the quadratic form
f (x 1, x 2, x 3) \u003d 2x 1 2 + 4x 1 x 2 - 3x 2 2 - x 2 x 3.

To do this, first select the full square for the variable x 1:

f (x 1, x 2, x 3) \u003d 2 (x 1 2 + 2x 1 x 2 + x 2 2) - 2x 2 2 - 3x 2 2 - x 2 x 3 \u003d 2 (x 1 + x 2) 2 - 5x 2 2 - x 2 x 3.

Now we select the full square for the variable x 2:

f (x 1, x 2, x 3) \u003d 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 =
\u003d 2 (x 1 + x 2) 2 - 5 (x 2 - (1/10) x 3) 2 - (1/20) x 3 2.

Then a non-degenerate linear transformation y 1 \u003d x 1 + x 2, y 2 \u003d x 2 - (1/10) x 3 and y 3 \u003d x 3 brings this quadratic form to the canonical form f (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 defined ambiguously (the same quadratic form can be reduced to the canonical form different ways). However, the different ways canonical forms have a number common properties. In particular, the number of terms with positive (negative) coefficients of a quadratic form does not depend on how the form is reduced to this form (for example, in the considered example there will always be two negative and one positive coefficient). This property is called the law of inertia of quadratic forms.

Let us verify this by reducing the same quadratic form to the canonical form in a different way. Let's start the transformation with the variable x 2:
f (x 1, x 2, x 3) \u003d 2x 1 2 + 4x 1 x 2 - 3x 2 2 - x 2 x 3 \u003d -3x 2 2 - x 2 x 3 + 4x 1 x 2 + 2x 1 2 \u003d - 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 =
\u003d -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 \u003d f (y 1, y 2, y 3) = -3y 1 2 -
-3y 2 2 + 2y 3 2, where y 1 \u003d - (2/3) x 1 + x 2 - (1/6) x 3, y 2 \u003d (2/3) x 1 + (1/6) x 3 and y 3 = x 1 . Here, a positive coefficient 2 for y 3 and two negative coefficients (-3) for y 1 and y 2 (and using another method, we got a positive coefficient 2 for y 1 and two negative coefficients - (-5) for y 2 and (-1 /20) for y 3).

It should also be noted that the rank of a matrix of a quadratic form, called the rank of the quadratic form, is equal to the number of non-zero 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) \u003d x 1 2 + x 2 2 is positive definite, because is the sum of squares, and the quadratic form f 2 (X) \u003d -x 1 2 + 2x 1 x 2 - x 2 2 is negative definite, because represents it can be represented as f 2 (X) \u003d - (x 1 - x 2) 2.

In most practical situations, it is somewhat more difficult to establish the sign-definiteness of a quadratic form, so one of the following theorems is used for this (we formulate them without proofs).

Theorem. A quadratic form is positive (negative) definite if and only if all eigenvalues its matrices are positive (negative).

Theorem (Sylvester's criterion). A quadratic form is positive definite if and only if all principal minors of the matrix of this form are positive.

Major (corner) minor The k-th order of the matrix A of the n-th order is 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, we examine the quadratic form f (x 1, x 2) = 2x 1 2 + 4x 1 x 2 + 3x 2 2 for sign-definiteness.

= (2 - l)*
*(3 - l) - 4 \u003d (6 - 2l - 3l + l 2) - 4 \u003d l 2 - 5l + 2 \u003d 0; D \u003d 25 - 8 \u003d 17;
. Therefore, the quadratic form is positive definite.

Method 2. The main minor of the first order of the matrix A D 1 = a 11 = 2 > 0. The main minor of the second order D 2 = = 6 - 4 = 2 > 0. Therefore, according to the Sylvester criterion, the quadratic form is positive definite.

We examine another quadratic form for sign-definiteness, f (x 1, x 2) \u003d -2x 1 2 + 4x 1 x 2 - 3x 2 2.

Method 1. Let's construct a matrix of quadratic form А = . The characteristic equation will have the form = (-2 - l)*
*(-3 - l) - 4 \u003d (6 + 2l + 3l + l 2) - 4 \u003d l 2 + 5l + 2 \u003d 0; D \u003d 25 - 8 \u003d 17;
. Therefore, the quadratic form is negative definite.

Service assignment. An online calculator is used to find Hessian matrices and determining the type of function (convex or concave) (see example). The decision is made in Word format. For a function of one variable f(x), the intervals of convexity and concavity are determined.

Function entry rules:

A twice continuously differentiable function f(x) is convex (concave) if and only if Hessian matrix the function f(x) in x is positively (negatively) semi-definite for all x (see points of local extrema of functions of several variables).

Critical points of the function:

if the Hessian is positively defined, then x 0 is a local minimum point of the function f(x) ,
if the Hessian is negative definite, then x 0 is the local maximum point of the function f(x),
if the Hessian is not sign-definite (takes both positive and negative values) and non-degenerate (det G(f) ≠ 0), then x 0 is a saddle point of the function f(x).

Criteria for the certainty of a matrix (Sylvester's theorem)

positive definiteness:

all diagonal elements of the matrix must be positive;
all leading principal determinants must be positive.

For positive semidefinite matrices Sylvester's criterion sounds like this: A form is positive semidefinite if and only if all principal minors are non-negative. If the Hessian matrix at a point is positive semi-definite (all major minors are non-negative), then this is a minimum point (however, if the Hessian is semi-definite and one of the minors is 0, then this may also be a saddle point. Additional checks are needed).

Positive semidefiniteness:

all diagonal elements are non-negative;
all principal determinants are non-negative.

The main determinant is the main minor determinant.

A square symmetric matrix of order n whose elements are partial derivatives of the second order objective function, called the Hessian matrix and is denoted:

For a symmetric matrix to be positive definite, it is necessary and sufficient that all its diagonal minors be positive, i.e.

for the matrix A = (a ij) are positive.

Negative certainty.
For a symmetric matrix to be negative definite, it is necessary and sufficient that the inequalities take place:
(-1) k D k > 0, k=1,.., n.
In other words, in order for the quadratic form to be negative definite, it is necessary and sufficient that the signs of the angular minors of the matrix of the quadratic form alternate, starting with the minus sign. For example, for two variables, D 1< 0, D 2 > 0.

If the Hessian is semi-definite, then it can also be an inflection point. Additional studies are needed, which can be carried out according to one of the following options:

Downgrading. A change of variables is made. For example, for a function of two variables, this is y=x , as a result, we get a function of one variable x . Next, the behavior of the function on the lines y=x and y=-x is investigated. If in the first case the function at the point under study will have a minimum, and in the other case a maximum (or vice versa), then the point under study is a saddle point.
Finding the eigenvalues of the Hessian. If all values are positive, the function at the point under study has a minimum, if all are negative, there is a maximum.
Investigation of the function f(x) in a neighborhood of the point ε. The variables x are replaced by x 0 +ε. Further, it is necessary to prove that the function f(x 0 + ε) of one variable ε is either greater than zero (then x 0 is the minimum point), or less than zero(then x 0 is the maximum point).

Note. To find inverse hessian it is enough to find the inverse matrix .

Example #1. Which of the following functions are convex or concave: f(x) = 8x 1 2 +4x 1 x 2 +5x 2 2 .
Solution. 1. Let's find partial derivatives.

2. Let's solve the system of equations.
-4x1 +4x2 +2 = 0
4x1 -6x2 +6 = 0
We get:
a) Express x 1 from the first equation and substitute it into the second equation:
x2 = x2 + 1/2
-2x 2 +8 = 0
Where x 2 \u003d 4
These x 2 values are substituted into the expression for x 1 . We get: x 1 \u003d 9 / 2
The number of critical points is 1.
M 1 (9 / 2 ;4)
3. Find the partial derivatives of the second order.

4. Calculate the value of these partial derivatives of the second order at the critical points M(x 0 ;y 0).
Calculate values for point M 1 (9 / 2 ;4)

We build the Hessian matrix:

D 1 = a 11< 0, D 2 = 8 > 0
Since the diagonal minors have different signs, nothing can be said about the convexity or concavity of the function.

positive definite quadratic forms

Definition. Quadratic form from n unknown is called positive definite, if its rank is equal to the positive index of inertia and is equal to the number of unknowns.

Theorem. A quadratic form is positive definite if and only if, on any nonzero set of variable values, it takes positive values.

Proof. Let the quadratic form be a non-degenerate linear transformation of the unknowns

returned to normal

For any non-zero set of variable values, at least one of the numbers different from zero, i.e. . The necessity of the theorem is proved.

Assume that the quadratic form takes positive values on any non-zero set of variables, but its index of inertia is positive. By a non-degenerate linear transformation of the unknowns

Let's bring it back to normal. Without loss of generality, we can assume that in this normal form the square of the last variable is either absent or enters it with a minus sign, i.e. , where or . Suppose that is a non-zero set of values of variables obtained as a result of solving the system linear equations

In this system, the number of equations is equal to the number of variables and the determinant of the system is nonzero. By Cramer's theorem, the system has a unique solution, and it is nonzero. For this set. Contradiction with the condition. We arrive at a contradiction with the assumption, which proves the sufficiency of the theorem.

Using this criterion, it is not possible to determine from the coefficients whether a quadratic form is positive-definite. The answer to this question is given by another theorem, for the formulation of which we introduce one more concept. Principal Diagonal Matrix Minors are the minors located in its upper left corner:

, , , … , .

Theorem.A quadratic form is positive definite if and only if all its principal diagonal minors are positive.

Proof we will carry out the method of full mathematical induction by number n quadratic form variables f.

Hypothesis of induction. Assume that for quadratic forms with fewer variables n the statement is correct.

Consider the quadratic form from n variables. Collect in one bracket all the terms containing . The remaining terms form a quadratic form in variables. By the induction hypothesis, the statement is true for it.

Assume that the quadratic form is positive definite. Then the quadratic form is also positive definite. If we assume that this is not the case, then there is a non-zero set of variable values , for which and correspondingly, , which contradicts the fact that the quadratic form is positive definite. By the induction hypothesis, all principal diagonal minors of a quadratic form are positive, i.e. all first principal minors of a quadratic form f are positive. Last principal minor of a quadratic form – is the determinant of its matrix. This determinant is positive, since its sign coincides with the sign of the matrix of its normal form, i.e. with the sign of the identity matrix determinant.

Let all principal diagonal minors of the quadratic form be positive. Then all principal diagonal minors of the quadratic form are positive from the equality . By the induction hypothesis, the quadratic form is positive definite, so there is a non-degenerate linear transformation of variables that reduces the form to the form of the sum of squares of new variables . This linear transformation can be extended to a nondegenerate linear transformation of all variables by setting . The quadratic form is reduced by this transformation to the form

The matrix A, composed of these coefficients, is called the quadratic form matrix. It's always symmetrical matrix (i.e. a matrix symmetric with respect to the main diagonal, a ij = a ji).

In matrix notation, the quadratic form has the form f(X) = X T AX, where

Indeed

For example, let's write the quadratic form in matrix form.

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-degenerate matrix of order n. 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 the 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 a linear transformation.

The quadratic form is called canonical(It has canonical view), if all its coefficients a ij \u003d 0 at i≠j, i.e. f (x 1, x 2,..., x n) \u003d a 11 x 1 2 + a 22 x 2 2 + ... + a nn x n 2 = .

Its matrix is diagonal.

Theorem(the proof is not given here). Any quadratic form can be reduced to a canonical form using a non-degenerate linear transformation.

For example, let's bring to the 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, first select the full square for the variable x 1:

f (x 1, x 2, x 3) \u003d 2 (x 1 2 + 2x 1 x 2 + x 2 2) - 2x 2 2 - 3x 2 2 - x 2 x 3 \u003d 2 (x 1 + x 2) 2 - 5x 2 2 - x 2 x 3.

Now we select the full square for the variable x 2:

f (x 1, x 2, x 3) \u003d 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 \u003d \u003d 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 \u003d x 1 + x 2, y 2 \u003d x 2 - (1/10) x 3 and y 3 \u003d x 3 brings this quadratic form to the canonical form f (y 1, y 2, y 3) \u003d 2y 1 2 - 5y 2 2 - (1/20)y 3 2 .

Note that the canonical form of a quadratic form is defined ambiguously (the same quadratic form can be reduced to the canonical form in different ways1). 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 how the form is reduced to this form (for example, in the considered example there will always be two negative and one positive coefficient). This property is called the law of inertia of quadratic forms.

Let us verify this by reducing the same quadratic form to the canonical form in a different way. Let's start the transformation with the variable x 2: f (x 1, x 2, x 3) \u003d 2x 1 2 + 4x 1 x 2 - 3x 2 2 - x 2 x 3 \u003d -3x 2 2 - x 2 x 3 + 4x 1 x 2 + 2x 1 2 \u003d -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 \u003d f (y 1, y 2, y 3) \u003d -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, a positive coefficient 2 at y 3 and two negative coefficients (-3) at y 1 and y 2 (and using another method, we got a positive coefficient 2 at y 1 and two negative coefficients - (-5) at y 2 and (-1/20) at y 3 ).

The quadratic form f(X) is called positively(negative)certain, if for all values of variables that are not simultaneously equal to zero, it is positive, i.e. f(X) > 0 (negative, i.e. f(X)< 0).

Theorem. A quadratic form is positive (negative) definite if and only if all eigenvalues of its matrix are positive (negative).

Theorem (Sylvester's criterion). A quadratic form is positive definite if and only if all principal minors of the matrix of this form are positive.

Major (corner) minor The k-th order of the An-th order matrix is 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, we 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. The main minor of the first order of the matrix A  1 = a 11 = 2 > 0. The main minor of the second order  2 = = 6 - 4 = 2 > 0. Therefore, according to the Sylvester criterion, the quadratic form is positive definite.

We examine another quadratic form for sign-definiteness, f (x 1, x 2) \u003d -2x 1 2 + 4x 1 x 2 - 3x 2 2.

Method 1. Let's construct a matrix of quadratic form А = . The characteristic equation will have the form = (-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. The main minor of the first order of the matrix A  1 \u003d a 11 \u003d \u003d -2< 0. Главный минор второго порядка 2 = = 6 – 4 = 2 >0. Therefore, according to the Sylvester criterion, the quadratic form is negative definite (the signs of the principal minors alternate, starting from minus).

And as another example, we examine the quadratic form f (x 1, x 2) \u003d 2x 1 2 + 4x 1 x 2 - 3x 2 2 for sign-definiteness.

Method 1. Let's construct a matrix of quadratic form А = . 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. Therefore, a quadratic form cannot be either negative or positive definite, i.e. this quadratic form is not sign-definite (it can take values of any sign).

Method 2. The main minor of the first order of the matrix A  1 = a 11 = 2 > 0. The main 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 a canonical form is convenient to use when non-zero coefficients occur under the squares of the variables. If they are not there, it is still possible to carry out the conversion, but you have to use some other tricks. 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 =

\u003d (x 1 + x 2) 2 - x 1 2 - x 2 2 \u003d (x 1 + x 2) 2 - (x 1 2 - 2x 1 x 2 + x 2 2) - 2x 1 x 2 \u003d (x 1 + x 2) 2 - - (x 1 - x 2) 2 - 2x 1 x 2; 4x 1 x 2 \u003d (x 1 + x 2) 2 - (x 1 - x 2) 2; f (x 1, x 2) \u003d 2x 1 x 2 \u003d (1/2) * * (x 1 + x 2 ) 2 - (1/2) * (x 1 - x 2) 2 \u003d f (y 1, y 2) \u003d (1/2) y 1 2 - (1/2) y 2 2, where y 1 \u003d x 1 + x 2, ay 2 \u003d x 1 - x 2.