"money" continues to monitor the profitability of financial instruments. Law of inertia of quadratic forms

Above the field K (\displaystyle K) And e 1 , e 2 , … , e n (\displaystyle e_(1),e_(2),\dots ,e_(n))- basis in L (\displaystyle L).

• A quadratic form is positive definite if and only if all angular minors of its matrix are strictly positive.
• A quadratic form is negative definite if and only if the signs of all angular minors of its matrix alternate, and the minor of order 1 is negative.

A bilinear form polar to a positive definite quadratic form satisfies all the dot product axioms.

Canonical view

Real case

In case K = R (\displaystyle K=\mathbb (R) )(field of real numbers), for any quadratic form there is a basis in which its matrix is ​​diagonal, and the form itself has canonical view(normal view):

Q (x) = x 1 2 + ⋯ + x p 2 − x p + 1 2 − ⋯ − x p + q 2 , 0 ≤ p , q ≤ r , p + q = r , (∗) (\displaystyle Q(x) =x_(1)^(2)+\cdots +x_(p)^(2)-x_(p+1)^(2)-\cdots -x_(p+q)^(2),\quad \ 0\leq p,q\leq r,\quad p+q=r,\qquad (*))

Where r (\displaystyle r)- rank of the quadratic form. In the case of a non-degenerate quadratic form p + q = n (\displaystyle p+q=n), and in the case of degenerate - p+q< n {\displaystyle p+q.

To reduce a quadratic form to canonical form, the Lagrange method or orthogonal basis transformations are usually used, and a given quadratic form can be brought to canonical form in more than one way.

Number q (\displaystyle q)(negative terms) is called inertia index given quadratic form, and the number p − q (\displaystyle p-q)(the difference between the number of positive and negative terms) is called signature quadratic form. Note that sometimes the signature of a quadratic form is the pair (p , q) (\displaystyle (p,q)). Numbers p , q , p − q (\displaystyle p,q,p-q) are invariants of quadratic form, i.e. do not depend on the method of reducing it to canonical form ( Sylvester's law of inertia).

Complex case

In case K = C (\displaystyle K=\mathbb (C) )(field of complex numbers), for any quadratic form there is a basis in which the form has the canonical form

Q (x) = x 1 2 + ⋯ + x r 2 , (∗ ∗) (\displaystyle Q(x)=x_(1)^(2)+\cdots +x_(r)^(2),\qquad ( **))

Where r (\displaystyle r)- rank of the quadratic form. Thus, in the complex case (as opposed to the real case), the quadratic form has one single invariant - rank, and all non-degenerate forms have the same canonical form (sum of squares).

Normal view of a quadratic form.

According to Lagrange's theorem, any quadratic form can be reduced to canonical form. That is, there is a diagonalizing (canonical) basis in which the matrix of this quadratic form has a diagonal form

Where . Then in this basis the quadratic form has the form

Let there be positive and negative elements among the non-zero elements, and . By changing, if necessary, the numbering of the basis vectors, you can always ensure that in a diagonal matrix of a quadratic form the first elements are positive, the rest are negative (if , then the last elements in the matrix are zeros). As a result, the quadratic form (10.17) can be written in the following form

As a result of replacing variables with variables according to the system:

the quadratic form (6.18) will take a diagonal form, in which the coefficients of the squares of the variables are one, minus one or zero:

where the matrix of quadratic form (10.19) has the diagonal form

Definition 10.9. Recording (10.19) is called normal looking quadratic form, and the diagonalizing basis in which the quadratic form has matrix (10.20) is called normalizing basis.

Thus, in the normal form (10.19) of the quadratic form, the diagonal elements of the matrix (10.20) can be ones, minus ones or zeros, and they are located so that first there are ones, then minus ones, then zeros (cases of turning to zero are not excluded specified values ​​, , ).

Thus, the following theorem is proven.

Theorem 10.3. Any quadratic form can be reduced to normal form (10.19) with a diagonal matrix (10.20).

Law of inertia quadratic form

The quadratic form can be reduced to canonical form in various ways (Lagrange method, orthogonal transformation method or Jacobi method). But, despite the variety of canonical forms for a given quadratic form, there are characteristics of its coefficients that remain unchanged in all these canonical forms. We are talking about the so-called numerical invariants quadratic form. One of the numerical invariants of a quadratic form is the rank of the quadratic form.

Theorem 10.4 ( on the rank invariance of a quadratic form ) The rank of a quadratic form does not change under non-degenerate linear transformations and is equal to the number of non-zero coefficients in any of its canonical forms. In other words, the rank of a quadratic form is equal to the number of non-zero eigenvalues ​​of the matrix of the quadratic form (taking into account their multiplicity).

Definition 10.10. The rank of a quadratic form is called inertia index. The number of positive and the number () of negative numbers in normal form (3) of the quadratic form are called positive And negative indices inertia of quadratic form, respectively. In this case the list is called signature quadratic form.

Positive and negative inertia indices are numerical invariants of the quadratic form. The theorem called law of inertia.

Theorem 10.5 ( law of inertia ) .The canonical form (10.17) of the quadratic form is uniquely defined, that is, the signature does not depend on the choice of the diagonalizing basis (does not depend on the method of reducing the quadratic form to the canonical form).

□ The statement of the theorem means that if the same quadratic form using two non-singular linear transformations

reduced to various canonical forms ():

then it is mandatory, that is, the number of positive coefficients coincides with the number of positive coefficients.

Contrary to the statement, suppose that . Since the transformations (10.21) are non-degenerate, we express the canonical variables from them:

Let's find a vector such that the corresponding vectors have the form

To do this, we present the matrices in the following block forms:

where the denotations are -matrix, -matrix, -matrix, -matrix.

As a result of block representations of matrices and, we will compose a homogeneous system of linear algebraic equations, taking the first equations from (10.22), and the last equations from (10.23):

The resulting system contains equations and unknowns (vector component). Since , then, that is, in this system the number of equations is less than the number of unknowns, and it has an infinite number of solutions, among which a non-zero solution can be identified.

On the resulting vector, the shape values ​​have different signs:

which is impossible. This means that the assumption that is false, that is, .

From what it follows that the signature does not depend on the choice of the diagonalizing basis. ■

As an illustration of the law of inertia, it can be shown that the quadratic form of three variables is:

two non-singular linear transformations, with corresponding matrices

(the first matrix corresponds to the Lagrange method, the second – to the method of orthogonal transformations) is reduced, respectively, to two different canonical forms

Moreover, both canonical forms have the same signature

6. Definite and alternating quadratic forms

Quadratic forms are divided into types depending on the set of values ​​they accept.

Definition 10.11. The quadratic form is called:

positive definite

negatively defined, if for any non-zero vector: ;

non-positive definite (negative semi-definite), if for any non-zero vector: ;

non-negative definite (positive semi-definite), if for any non-zero vector: ;

alternating sign, if there are nonzero vectors , : .

Definition 10.12. Positively (negatively) definite quadratic forms are called definite. Non-positive (non-negative) definite quadratic forms are called constant sign.

The type of a quadratic form can be easily determined by reducing it to canonical (or normal) form. The following two theorems are true.

Theorem 10.6. Let the quadratic form be reduced to canonical form and have signature ( , ). Then:

Is positive definite ;

Is negatively defined ;

Is non-positive definite ;

Is non-negative definite ;

Is alternating sign.). Then: non-negative definite for all ;

Is alternating sign among the eigenvalues ​​there are both positive and negative.

It has been established that the number of non-zero canonical coefficients of a quadratic form is equal to its rank and does not depend on the choice of a non-degenerate transformation with the help of which the form A(x, x) is reduced to canonical form. In fact, the number of positive and negative coefficients does not change either.

Theorem11.3 (law of inertia of quadratic forms). The number of positive and negative coefficients in the normal form of a quadratic form does not depend on the method of reducing the quadratic form to normal form.

Let the quadratic form f rank r from n unknown x 1 , x 2 , …, x n reduced to normal form in two ways, that is

f = + + … + –
– … – ,

f = + + … + – – … – . It can be proven that k = l.

Definition 11.14. The number of positive squares in the normal form to which the real quadratic form is reduced is called positive inertia index this form; number of negative squares – negative inertia index, and their sum is inertia index quadratic form or signature forms f.

If p– positive inertia index; q– negative inertia index; k = r = p + q– inertia index.

Classification of quadratic forms

Let the quadratic form A(x, x) the inertia index is equal to k, positive inertia index is equal to p, negative inertia index is equal to q, Then k = p + q.

It has been proven that in any canonical basis f = {f 1 , f 2 , …, f n) this quadratic form A(x, x) can be reduced to normal form A(x, x) = + + … + –
– … – , Where  1 ,  2 , …,  n vector coordinates x in basis ( f}.

Necessary and sufficient condition for the sign of a quadratic form

Statement11.1. A(x, x), specified in n V, was definite sign, it is necessary and sufficient that either a positive inertia index p, or negative inertia index q, was equal to the dimension n space V.

Moreover, if p = n, then the form positively x ≠ 0 A(x, x) > 0).

If q = n, then the form negative defined (that is, for any x ≠ 0 A(x, x) < 0).

Necessary and sufficient condition for the alternation of signs of a quadratic form

Statement 11.2. In order for the quadratic form A(x, x), specified in n-dimensional vector space V, was alternating sign(that is, there are such x, y What A(x, x) > 0 and A(y, y) < 0) необходимо и достаточно, чтобы как положительный, так и отрицательный индексы инерции этой формы были отличны от нуля.

Necessary and sufficient condition for quasi-alternating quadratic form

Statement 11.3. In order for the quadratic form A(x, x), specified in n-dimensional vector space V, was quasi-alternating(that is, for any vector x or A(x, x) ≥ 0 or A(x, x) ≤ 0 and there is such a non-zero vector x, What A(x, x) = 0) it is necessary and sufficient for one of two relations to be satisfied: p < n, q= 0 or p = 0, q < n.

Comment. In order to apply these characteristics, the quadratic form must be reduced to canonical form. Sylvester's sign determination criterion 15 does not require this.

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

Concept of quadratic form: a function on a vector space defined by a homogeneous polynomial of the second degree in the coordinates of the vector.

Quadratic form from n unknowns is a 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 a matrix of quadratic form in a given basis. If the field characteristic is not equal to 2, we can assume that the matrix of quadratic form is symmetric, that is.

Write a matrix of quadratic form:

Hence,

In vector matrix form, the quadratic form is:

Canonical form of quadratic form: 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.

Lagrange method : sequential selection of complete squares. For example, if

Then a similar procedure is performed with the quadratic form, etc. If everything is not in the quadratic form, then after a preliminary transformation the matter comes down to the procedure considered. So, if, for example, then we assume

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

Rank, index and signature of quadratic form: Rank of quadratic form A is called the rank of the matrix A. The rank of a quadratic form does not change under non-degenerate transformations of unknowns.

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

The number of positive terms in 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 the positive and negative indices is called the signature of the quadratic form

Positive definite quadratic form: A real quadratic form is called positive definite (negative definite) if, for any real values ​​of the variables that are not simultaneously zero,

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 (resp. non-positive) forms.

Quadrics: 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), the general equation of a quadric is

This equation can be rewritten more compactly in matrix notation:

where x = ( x 1 , x 2 , x n+1 ) — row vector, x T is a transposed vector, Q— size matrix ( n+1)×( n+1) (it is assumed that at least one of its elements is non-zero), P is a row vector, and R— constant. Quadrics over real or complex numbers are most often considered. 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 a (affine or projective) algebraic variety of the second degree and codimension 1.

Transformations of plane and space.

Definition of plane transformation. Motion detection. properties of movement. Two types of movements: movement of the first kind and movement of the second kind. Examples of movements. Analytical expression of motion. Classification of plane movements (depending on the presence of fixed points and invariant lines). Group of plane movements.

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

Motion Definition: These are shape transformations that preserve the distances between points. If two figures are precisely aligned with each other through 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. When moving, points lying on a straight line transform into points lying on a straight line, and the order of their relative positions is maintained. When moving, the angles between 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 by 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 movements of the first kind is a movement of the first kind.

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

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

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

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

Rotation is a movement because it represents a mapping of the plane onto itself, in which distances are preserved.

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

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

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

Example: With 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 invariant line is a straight line - the axis of symmetry is a straight line of invariant points.

Theorem: If a movement does not have a single invariant point, then it has at least one invariant direction.

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

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

Therefore, based on 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 line) all straight lines

Plane motion group: In geometry, groups of self-compositions of figures play an important role. If is a certain figure on a plane (or in space), then we can consider the set of all those movements of the plane (or space) during which the figure turns into itself.

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

They are shown in Fig. 1 with red lines. The elements of the group of self-alignments of a regular triangle can be specified differently. To explain this, let us number the vertices of a regular triangle with the numbers 1, 2, 3. Any self-alignment of the triangle takes points 1, 2, 3 to the same points, but taken in a different order, i.e. can be conditionally written in the form of one of these brackets:

where the numbers 1, 2, 3 indicate the numbers of those vertices into which vertices 1, 2, 3 go as a result of the movement under consideration.

Projective spaces and their models.

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

The concept of projective space and the model of projective space:

Projective space over a field is a space consisting of lines (one-dimensional subspaces) of some linear space over a given field. Direct spaces are called dots projective space. This definition can be generalized to an arbitrary body

If it has dimension , then the dimension of the projective space is called number , and the projective space itself is denoted and 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 space.

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. The main feature of projective geometry is the principle of duality, which adds elegant symmetry to many designs. Projective geometry can be studied both from a purely geometric point of view, and from an analytical (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 considered to be the Euclidean plane with the addition of "line at infinity".

Whereas the properties of figures with which Euclidean geometry deals 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 through motion while preserving metric properties), there are more “deep-lying” properties of geometric figures that are preserved under transformations of a more general type than motion. Projective geometry deals with the study of properties of figures that are invariant under the class projective transformations, as well as these transformations themselves.

Projective geometry complements Euclidean geometry by providing beautiful and simple solutions to 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, complementation of Euclidean geometry, and structure over a field.

Axiomatization

Projective space can be defined using a different set of axioms.

Coxeter provides the following:

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

2. Each line has at least three points.

3. Through two points you can draw exactly one straight line.

4. If A, B, C, And D- various points and AB And CD intersect, then A.C. And BD intersect.

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

6. Two different planes intersect at least two points.

7. The three diagonal points of a complete quadrilateral are not collinear.

8. If three points are on a line X X

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

1. Through two points you can draw exactly one straight line.

2. Any two lines intersect.

3. There are four points, of which three are not collinear.

4. The three diagonal points of complete quadrilaterals are not collinear.

5. If three points are on a line X are invariant with respect to the projectivity of φ, then all points on X 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 proven without introducing an ideal point and line.

Extended plane - projective plane model: In the affine space A3 we take a bundle of lines S(O) with center at the point O and a plane Π that does not pass 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 define a mapping of the set of points of the plane Π onto the set of straight lines of the connective S (Fuck, pray if you got this question, forgive me)

Extended three-dimensional affine or Euclidean space—a model of projective space:

In order to make the mapping surjective, we repeat the process of formally extending the affine plane Π to the projective plane, Π, supplementing the plane Π with a set of improper points (M∞) such that: ((M∞)) = P0(O). Since in the map 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∞), represents 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 henceforth we will understand the last equality P0(O) = Π0 in the sense of equality of sets of points, but endowed with a different structure. By supplementing the affine plane with an improper line, we ensured that mapping (I.21) became bijective on the set of all points of the extended plane:

Images of flat and spatial figures during parallel design:

In stereometry, spatial figures are studied, but in the drawing they are depicted as flat figures. How should a spatial figure be depicted on a plane? Typically 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 by parallel projection A the point of intersection of the line is considered to be on the plane p l with plane p.

Thus, each point A space its projection is compared A" onto the plane p. This correspondence is called 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 a plane. Examples of projective transformations of the plane. Properties of projective transformations. Homology, properties of homology. Group of projective transformations.

The concept of projective transformation of a plane: The concept of a projective transformation generalizes the concept of a central projection. If we perform a central projection of the plane α onto some plane α 1, then a 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; Parallel projections can also be included in such a chain.

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

Properties of projective transformations:

During a projective transformation, three points not lying on a line are transformed into three points not lying on a line.

During a projective transformation, the frame turns into a frame.

During a projective transformation, a line goes into a straight line, and a pencil goes into a pencil.

Homology, properties of homology:

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

1. A line passing through non-coinciding corresponding homology points is an invariant line;

2. Lines passing through non-coinciding corresponding homology points belong to the same pencil, the center of which is an invariant point.

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

Group of projective transformations: consider the projective mapping of the projective plane P 2 onto itself, that is, the 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 sequential 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 with respect to the composition of projective transformations.

Law of inertia of quadratic forms. We have already noted above that the rank of a quadratic form is equal to the number of nonzero canonical coefficients. Thus, the number of non-zero canonical coefficients does not depend on the choice of non-degenerate transformation with the help of which the form is reduced to canonical form. In fact, with any method of reducing a form to canonical form, the number of positive and negative canonical coefficients does not change. This property is called the law of inertia of quadratic forms.

Let the form in the basis be determined by the matrix:

, (4.20)

where are the coordinates of the vector in the basis e. Let us assume that this form is reduced to the canonical form using a non-degenerate coordinate transformation

and are non-zero canonical coefficients, numbered so that the first of these coefficients are positive, and the following coefficients are negative:

, , …, , , …, .

Consider the following non-degenerate coordinate transformation:

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

called the normal form of a quadratic form.

Theorem 4.5 (law of inertia of quadratic forms). The number of terms with positive (negative) coefficients in the normal form of a quadratic form does not depend on the method of reducing the form to this form.

Consequence. Two quadratic forms are equivalent if and only if the ranks of the forms are equal and the positive and negative inertia indices coincide.

Classification of quadratic forms. In this section, using the concepts of inertia index, positive and negative inertia indices of a quadratic form, we will indicate how one can find out whether a quadratic form belongs to one or another of the types listed above (positive definite, negative definite, alternating and quasi-sign definite). In this case, we will call the inertia index of a quadratic form the number of non-zero canonical coefficients of this form (i.e. its rank), the positive inertia index the number of positive canonical coefficients, the negative inertia index the number of negative canonical coefficients. It is clear that the sum of the positive and negative inertia indices is equal to the inertia index. Negative and positive inertia indices are related by the relation, and the pair or is called signature quadratic form.

So, let the inertia index, positive and negative inertia indices of the quadratic form be equal to , and () respectively. In the previous paragraph it was proven that in any canonical basis this form can be reduced to the following normal form:

where are the coordinates of the vector in the basis.

Example 6 Find the normal form and signature of the quadratic form

The canonical form of this form is: . Let's put , , . Then . This is the normal form of a quadratic form. Positive inertia index: , negative inertia index. Therefore, the signature of the quadratic form is .

Theorem 4.6 (necessary and sufficient condition for the sign of a quadratic form) In order for a quadratic form given in an n-dimensional linear space L to be sign-definite, it is necessary and sufficient that either the positive index of inertia or the negative index of inertia is equal to the dimension of the space L. Moreover, if , then the form is positive definite, if , then the form is negative definite.

Comment. To clarify the issue of the definite sign of a quadratic form using the indicated criterion, we must bring this form to its canonical form.

Theorem 4.7 (necessary and sufficient condition for the alternation of signs of a quadratic form) In order for a quadratic form to be alternating, it is necessary and sufficient that both the positive and negative indices of inertia of this form are different from zero.

Theorem 4.8 (necessary and sufficient condition for the quasi-sign definiteness of a quadratic form) In order for a form to be quasi-sign definite, it is necessary and sufficient that the following relations hold: either , , or , .

Sylvester's criterion for the definite sign of a quadratic form. Let the form in the basis be determined by the matrix: and let , , …, be the angular (principal) minors and the determinant of the matrix . The following statement is true:

Theorem 4.9 (Sylvester criterion) In order for a quadratic form to be positive definite, it is necessary and sufficient that the inequalities , , …, .

In order for a quadratic form to be negative definite, it is necessary and sufficient that the signs of the angular minors alternate, and .

Corollary 1 In order for a quadratic form to be negative definite, it is necessary and sufficient that all angular minors of even order are positive and all angular minors of odd order are negative, or, otherwise, the inequalities , , ..., , are satisfied.

Corollary 2 In order for a quadratic form to be non-negative, it is necessary and sufficient that all major (not just angular) minors of its matrix are non-negative.

Corollary 3 In order for a quadratic form to be non-positive, it is necessary and sufficient that all leading minors of even order are non-negative and all leading minors of odd order are non-positive.

Corollary 4 In order for a quadratic form to be indefinite (alternating), it is necessary and sufficient that its matrix has a negative leading minor of even order and two leading minors of odd orders of different signs.

Example 7 Investigate quadratic forms for sign definiteness:

1) For a matrix of quadratic form, find all angular minors

In this case, again, it is impossible to give an answer only by the values ​​of the angular minors. Let's find all the major minors. The first order non-angular principal minors are 2 and 4. The second order non-angular principal minors are , . There is a negative leading minor of even order. Therefore, the quadratic form is indeterminate.