Sylvester's law of inertia. Quadratic Forms Theorem of Inertia for Quadratic Forms
September has proven to be a successful month for all asset classes. According to Deng's estimates, almost all investments have yielded positive results. At the same time, the highest income was brought by investments in gold, which benefited not only from the growth in the cost of the precious metal, but also from the weakening of the ruble. High returns were brought to investors by the main categories of mutual funds, deposits, as well as most of the Russian shares. Bond funds, popular in recent years, have become unprofitable, as well as Sberbank shares, which may suffer the most in the event of tougher US sanctions.
Vitaly Kapitonov
Five months later, gold was the most profitable investment of the month. According to Deng's estimates, having invested 100 thousand rubles in the precious metal on August 15, the investor could receive almost 5 thousand rubles in a month. income. This is the second highest monthly result this year. The investor could earn more in April - 9.3 thousand rubles.
The high profitability of investments in the precious metal is only partly due to the increase in its price. Since mid-August, the price of gold has increased by 2.4%, to $ 1205 per troy ounce. This was a reflection of US inflationary expectations. According to the US Department of Commerce, inflation in the country has slowed from 2.9% in July to 2.7% in August, but remains above the Fed's targets. Thus, inflation continues to rise, which will allow the Fed to raise the rate without sharp changes. The precious metal was supported by news that the US and Canadian authorities continue to try to find a compromise on a new NAFTA agreement. "The news eases trade concerns that have been weighing on the gold market and supporting the dollar," said Mikhail Sheibe, commodity strategist at Sberbank Investment Research. The effect of rising gold prices was enhanced by the growth of the dollar in Russia (+ 2.5%). As a result, ruble investments in the precious metal brought significant income.
However, further investments in gold should be treated with caution, according to market participants. The escalation of the trade confrontation between the US and China remains the key risk for investment in the precious metal. "The factor of political pressure has been ruled out, which means that the emergence of new barriers is practically a thing of the past. This development of events is negative for gold, since the demand for the dollar as a protective asset will increase," says Mikhail Sheibe.
What income was brought by investments in gold (%)
Sources: Bloomberg, Reuters, Sberbank.
Mutual funds remain among the most profitable financial products, and individual products of management companies were able to provide margins exceeding the indicator of gold. In October, the most successful investments were in industry-specific equity funds focused on metallurgical, telecommunications and oil and gas companies. According to Deng's estimates based on Investfunds data, by the end of the month investments in such funds would bring private investors from 2.2 thousand rubles to 5.2 thousand rubles.
Other categories of funds also provided high earnings: index funds, mixed investments, Eurobonds. Funds of these categories could bring their investors from 200 rubles. up to 4 thousand rubles. for 100 thousand investments.
Bonded funds, beloved by private investors, brought a negative result. Funds in this category are conservative, so the losses of private investors were symbolic - up to 1,000 rubles. In such conditions, investors began to take profits in bond funds. According to Investfunds, retail investors withdrew RUB 4 billion from bond funds in August. They took out faster from funds of this category in December 2014. Then, against the background of the devaluation of the ruble and the rapid growth of rates on the domestic market, investors withdrew more than 4.5 billion rubles from the funds.
Investors partly use the freed up liquidity to buy more risky equity funds. The volume of funds invested in funds of this category in August exceeded 3.5 billion rubles, which is 500 million rubles. more attraction in July. The demand for risky strategies has been growing for the sixth month in a row, and the volume of investments is taking an increasing share of the total inflow to retail funds. Telecommunications and oil and gas funds are in greatest demand among investors.
What income was brought by investments in mutual funds (%)
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Sources: National League of Governors, Investfunds.
August outsiders - shares - rose to third place from the fourth rating of "Money". Over the past month, investments in the MICEX index would have brought retail investors 3.4 thousand rubles. At the same time, the beginning of the period under review did not bode well for such a high result. In the period from August 15 to 18, the MICEX index fell by 1.2%. However, the situation has improved since 24 August. In three weeks, the index jumped almost 5% and rose to the level of 2374 points. This is just 2 points below the all-time high set in March.
However, in September, many stock indices of developing and developed countries showed positive dynamics. According to Bloomberg estimates, the Russian indices rose in dollar terms by only 4.4%. Only Turkish indices showed stronger growth, rising by 5.9-6.3%. Among the indicators of developed countries, the Italian FTSE MIB became the leader, adding 3.4% over the month.
The strongest gains were made in the shares of ALROSA, Gazprom, MMC Norilsk Nickel and Magnit: on these securities the investor could earn 4.2-8.3 thousand rubles. for every hundred thousand investments. According to Anton Startsev, a leading analyst at Olma Investment Company, investors' interest in ALROSA shares was supported by the statement of Finance Minister Anton Siluanov that the company could use 75% of its net profit to pay dividends.
An exception to the overall picture was the shares of RusHydro, Rostelecom, Aeroflot, investments in which would have brought a loss of 200 rubles or more. up to 1.4 thousand rubles. The maximum losses would be for investors who have invested in Sberbank securities - 2.1 thousand rubles. His shares remain under pressure from comments from US State Department officials, who do not rule out the possibility of sanctions against the bank in November. Such prospects frighten international investors and force them to withdraw not only from OFZ, but also from the bank's securities.
After the collapse in August and September, Sberbank shares have become attractive for investment, analysts say. "A rebound in the securities of the largest Russian bank is very likely, and the risks of their purchases are quite justified. Mid-term investors should still focus on fixing profits at around RUB 180 per share," said Alexei Antonov, an analyst at ALOR Broker.
What income did investments in stocks bring (%)
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The concept of a quadratic form. Quadratic matrix. The canonical form of the quadratic form. Lagrange's method. Normal view of a quadratic form. The rank, index, and signature of the quadratic form. Positive definite quadratic form. Quadrics.
Quadratic form concept: function on vector space, given by a homogeneous polynomial of the second degree in the coordinates of the vector.
Quadratic from nunknowns 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 \u200b\u200bcalled 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 the quadratic form is symmetric, that is.
Write a matrix of quadratic form:
Hence,
In vector-matrix form, the quadratic form is:
The canonical form of the quadratic form: 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.
Lagrange method : sequential selection of complete squares. For example, if
Then a similar procedure is performed with a quadratic form, etc. If in a quadratic form everything is still there, then after a preliminary transformation the matter is reduced to the considered procedure. So, if, for example, then we put
Normal view of the quadratic form: A normal quadratic form is a canonical quadratic form in which all coefficients are +1 or -1.
Rank, index and signature of the quadratic form:By the rank of the quadratic form AND called the rank of the matrix AND... The rank of the quadratic form does not change under non-degenerate 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 a quadratic signature
Positive definite quadratic form: A real quadratic form is called positive definite (negative definite), if for any real values \u200b\u200bof variables that are not simultaneously equal to zero
In this case, the matrix is \u200b\u200balso 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.
Quadrics:Quadric - n-dimensional hypersurface in n+ 1-dimensional space, defined as the set of zeros of a second-degree polynomial. If you enter coordinates ( x 1 , x 2 , x n +1) (in Euclidean or affine space), the general equation of the quadric has the form
This equation can be rewritten more compactly in matrix notation:
where x \u003d ( x 1 , x 2 , x n +1) is a row vector, x T - transposed vector, Q - 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.
Define a plane transformation. Motion detection. properties of motion. Two types of movements: movement of the 1st kind and movement of the 2nd kind. Examples of movements. Analytical expression of movement. 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 by moving) the plane. The plane transformation is called affine, if it transforms any three points lying on one straight line into three points also lying on one straight line and at the same time preserves the simple ratio of three points.
Motion detection: it is a shape transformation that maintains the distance between points. If two figures are precisely aligned with each other by means of movement, then these figures are the same, equal.
Motion properties:any orientation-preserving motion of the plane is either a parallel translation or a rotation, any orientation-reversing movement of the plane is either axial symmetry or sliding symmetry. Points lying on a straight line pass into points lying on a straight line during movement, and the order of their relative position is preserved. When moving, the angles between the half-lines are preserved.
Two types of movements: movement of the 1st kind and movement of the 2nd 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 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 a given vector. Parallel transfer to the vector a is called 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 motion because it is a mapping of a plane onto itself that preserves distances. This movement can be visualized as a shift of the entire plane in the direction of a given vector a by its length.
Turn. We denote the point O ( pivot center) and set the angle α ( angle of rotation). The rotation of the plane around the point O by the angle α is called the mapping of the plane onto itself, in which each point M is mapped to the point M 1, that OM \u003d OM 1 and the angle MOM 1 is α. In this case, point O remains in place, that is, it is mapped into itself, and all other points rotate around point O in the same direction - clockwise or counterclockwise (the figure shows counterclockwise rotation).
Rotation is movement because it is a plane-to-self mapping that maintains distances.
Analytical expression of movement: analytical connection between the coordinates of the preimage and the image of a point has the form (1).
Classification of plane movements (depending on the presence of fixed points and invariant lines): Definition:
A point of the plane is invariant (fixed) if it transforms into itself under the given transformation.
Example: With central symmetry, the center point of symmetry is invariant. When turning, the turning center point is invariant. With axial symmetry, a straight line is invariant - the axis of symmetry is a straight line of invariant points.
Theorem: If the motion has no 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 is moving, the ray translates into itself, then this movement is either an identical transformation, or symmetry with respect to the straight line containing this 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 | not |
2. Identical transformation | all points of the plane | all straight |
3. Central symmetry | point 0 - center | all lines passing through point 0 |
4. Parallel transfer | not | all straight |
Type II movement. | ||
5. Axial symmetry. | set of points | axis of symmetry (straight line) all straight lines |
Group of plane movements: Self-alignment groups of figures play an important role in geometry. If - some figure on a plane (or in space), then you can consider the set of all those movements of the plane (or space), in which the figure goes into itself.
This multitude 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 at angles around a point and symmetries about three straight lines.
They are shown in fig. 1 with red lines. The elements of the self-alignment group of a regular triangle can be specified differently. To clarify this, let's number the vertices of a regular triangle with numbers 1, 2, 3. Any self-alignment of the triangle translates points 1, 2, 3 into the same points, but taken in a different order, ie. can be conventionally written in the form of one of these brackets:
where the numbers 1, 2, 3 denote the numbers of those vertices into which the vertices 1, 2, 3 as a result of the considered movement are transformed.
Projective spaces and their models.
The concept of projective space and models of projective space. Basic facts of projective geometry. A bundle of straight lines centered at the point O is a model of the projective plane. Projective points. Extended plane - a model of the projective plane. Extended three-dimensional affine or Euclidean space - a model of projective space. Images of planar and spatial figures in parallel design.
The concept of projective space and models of projective space:
A projective space over a field is a space consisting of straight lines (one-dimensional subspaces) of some linear space over a given field. Straight spaces are called dots projective space. This definition lends itself to generalization to an arbitrary body
If has dimension, then the dimension of the projective space is a 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 duality principle, which adds elegant symmetry to many designs. Projective geometry can be studied both from a purely geometric point of view, so from analytic (using homogeneous coordinates) and salgebraic, considering the projective plane as a structure over a field. Often, and historically, the real projective plane is viewed as the Euclidean plane with the addition of "a straight line at infinity".
Whereas the properties of the figures with which Euclidean geometry deals are metric (specific values \u200b\u200bof angles, segments, areas), and the equivalence of figures is equivalent to their congruence (ie, when figures can be translated into one another by means of movement with the preservation of metric properties), there are more "deep-lying" properties of geometric shapes that are preserved under transformations of a more general type than motion. Projective geometry studies the properties of figures that are invariant under the class projective transformationsas 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, complement to 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 line and a point is not on it.
2. Each line has at least three points.
3. Exactly one straight line can be drawn through two points.
4. If A, B, Cand D - various points and AB and CD intersect, then AC and BD intersect.
5. If ABC - plane, then there is at least one point not in the plane ABC.
6. Two different planes intersect at least two points.
7. Three diagonal points of a complete quadrilateral are not collinear.
8. If three points on a straight line X X
The projective plane (without the third dimension) is determined by slightly 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 three points on a straight line X are invariant with respect to the projectivity of φ, then all points on X are invariant with respect to φ.
6. Desargues's theorem: If two triangles are perspective through a point, then they are perspective through a line.
In the presence of the third dimension, Desargues's theorem can be proved without introducing ideal points and lines.
Extended plane - model of the projective plane: take in the affine space A3 a bundle of lines S (O) centered at the point O and a plane Π not passing through the center of the bundle: O 6∈ Π. A bunch of lines in an affine space is a model of the projective plane. Let us assign a mapping from the set of points of the plane Π to the set of straight lines of the connective S (Shit, pray if you got this question, sorry)
Extended three-dimensional affine or Euclidean space - a model of projective space:
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∞)) \u003d P0 (O). Since in the map the preimage of each plane of the bundle of planes S (O) is a straight line on the plane d, it is obvious that the set of all improper points of the extended plane: Π \u003d Π ∩ (M∞), (M∞), is an improper straight line d∞ of the extended the plane that is the inverse image of the singular plane Π0: (d∞) \u003d P0 (O) (\u003d Π0). (I.23) Let us agree that the last equality P0 (O) \u003d Π0 here and in what follows will be understood in the sense of equality of sets of points, but endowed with a different structure. By complementing the affine plane with an improper straight line, we have achieved that the mapping (I.21) becomes bijective on the set of all points of the extended plane:
Images of planar and spatial shapes in 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, geometry uses parallel design for this. Let p be some plane, l - a straight line intersecting it (Fig. 1). Through an arbitrary point Anot belonging to the straight line l, draw a straight line parallel to the straight line l... The point of intersection of this line with the plane p is called the parallel projection of the point A on the plane p in the direction of the straight line l... Let's denote it A". If point A belongs to direct l, then parallel projection Aon the plane p is the point of intersection of the straight line l with the plane p.
Thus, each point A space is compared to its projection A"on the plane p. This correspondence is called parallel projection on the plane p in the direction of the straight line l.
The group of projective transformations. Application to problem solving.
The concept of a projective transformation of a plane. Examples of projective transformations of the plane. Properties of projective transformations. Homology, properties of homology. The group of projective transformations.
The concept of a projective transformation of a plane: The concept of a projective transformation generalizes the concept 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; parallel projections can be included in such a chain.
Examples of projective transformations of the plane: 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 straight 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 goes over into itself.
Properties of projective transformations:
Under a projective transformation, three points that do not lie on the line go over to three points that do not lie on the line.
With a projective transformation, the frame is transformed into a frame.
Under a projective transformation, a straight line turns into a straight line, a bundle turns into a bundle.
Homology, homology properties:
A projective transformation of a plane that has a straight line of invariant points, and hence a pencil of invariant straight lines, is called homology.
1. The straight line passing through the non-coinciding corresponding points of the homology is an invariant straight line;
2. Lines passing through the corresponding non-coinciding points of the homology belong to one sheaf, the center of which is an invariant point.
3. The point, its image and the center of homology are collinear.
The 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 '\u003d 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 \u003d 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.
The quadratic form can be brought to the normal form by various non-degenerate linear transformations (coordinate transformations). The question arises: how are the various normal types of the same quadratic form related to each other?
Let be L n - n-dimensional linear space over the field R and let it be given a quadratic form j (and ). Let in L n the basis is given e = (e 1 , e 2, … , e n ) let it go ANDIs a matrix of a given form in this basis. Let be e 1 = (e 1 1 , e 2 1, … , e n 1 ) Is one of the bases in which j (and ) has a canonical form, and Tbasis transition matrix e to the base e 1 ... In the basis e 1 the form j (and ) has a diagonal matrix A 1... By the formula (56) A 1 \u003d T T × AND× T. Matrices Tand T T are nondegenerate. Matrix multiplication ANDto a non-degenerate matrix does not change the rank of the matrix ANDhence rang A\u003d rang A 1, i.e. in any basis, a matrix of quadratic form has the same rank.
Definition 63. Rank of a quadratic form defined on a linear space L n is called the rank of its matrix in any basis of this space.
Since the rank of a diagonal matrix is \u200b\u200bequal to the number of nonzero diagonal elements, any canonical form of a given quadratic form contains the same number of squares of variables with nonzero coefficients. This number is equal to the rank of the form. Therefore, the following statement is proved:
Theorem 66. A complex quadratic form is reduced by any nondegenerate linear transformation to the same normal form, consisting of rsquares of variables with unit coefficients, i.e. j= x 1 2 + x 2 2+ … + x r 2.
If the field R is a field of real numbers, then the normal form of the quadratic form will be j (and ) = x 1 2 + x 2 2 + … + x k 2 – x k + 1 2– … – x r 2.
Definition 64. The number of squares of variables included with a coefficient (+1) in the normal form of a real quadratic form is called positive index of inertia this form. The number of squares with a coefficient (–1) is called negative inertia index , the difference between the number of variables and the rank of the quadratic form (i.e. n -r) called it defect .
Theorem 67(law of inertia of quadratic forms ). The number of positive and the number of negative squares in the normal form, to which the quadratic form with real coefficients is reduced by a real non-degenerate linear transformation, does not depend on the choice of this transformation.
Evidence.Let be j (and ) Is a quadratic form given in the basis e = (e 1 , e 2, … , e n ) linear space L n over the field R , and = x 1 e 1 + x 2 e 2 + … + x n e n ... Let this form be reduced in two ways to two normal forms. According to previous results, both of these normal forms contain the same number of squares of variables with nonzero coefficients. Let be
j = y 1 2 + y 2 2 + … + y to 2 – y k + 1 2 – … – y r 2 =
\u003d z 1 2 + z 2 2 +… + z p 2 - z p + 1 2 -… - z r 2. (*)
Let be at i = , і = 1, 2, … , n (**), and z ј = , ј = 1, 2, … , n (***).
Since these formulas define non-degenerate transformations, their determinants are nonzero. It suffices to prove that k \u003d p.Let's pretend that to¹ r... Without loss of generality, we can assume that to< r... Let's compose the system of equations y 1 \u003d y 2 \u003d… \u003d y k \u003d z p + 1 \u003d… \u003d z r \u003d z r + 1 \u003d… \u003d z n \u003d0. This is the system n - p+ tolinear homogeneous equations in n unknown. Since the number of equations is less than the number of unknowns, it has nonzero solutions. Let be ( x 1 0, x 2 0, ..., x n 0) is one of them. Substituting this solution into formulas (**) and (***), we calculate all at i and z ј and substitute them into equality (*). We get - ( y k + 1 0) 2 – … – (y r 0) 2 = (z 1 0) 2 +(z 2 0) 2 + … +(z p 0) 2. This equality is possible if and only if y k + 1 0 = … = y r 0 = z 1 0 = z 2 0 = … = z p 0\u003d 0. We got that the system z 1 = z 2 = … = z p \u003d z p + 1 \u003d… \u003d z r \u003d z r + 1 \u003d… \u003d z n \u003d0 has a nonzero solution ( x 1 0, x 2 0, ..., x n 0), which is impossible, since the rank of this system is n... So our assumption is wrong. Hence, k \u003d p.
9.5. Positive definite quadratic forms
Definition 65. The real quadratic form is called positively defined if for any vector and ¹ 0 takes place j (and ) > 0.
Theorem 68. A real quadratic form is positive definite if and only if its rank and positive index of inertia are equal to the number of unknowns.
Evidence. Þ Let j (and ) Is a real positive definite quadratic form. Let it be reduced to normal form
y 1 2 + y 2 2 + … + y to 2 – y k + 1 2 – … – y r 2 (*),
in which either r< n or r \u003d nbut to< n ... Let the coordinate transformation, with the help of which the form is brought to normal form, be given by the formulas at i \u003d (**). The determinant of these formulas is nonzero. If a r< n, then let's take y 1 \u003d y 2 \u003d… \u003d y n – 1 = 0, at n \u003d 1 and substitute in (**). We get the system n linear inhomogeneous equations with nunknowns and with a determinant other than zero. According to Cramer's rule, this system has a unique solution. Obviously, this solution is not zero, therefore it defines a nonzero vector and ... But then j (and ) \u003d 0, which contradicts the definition of a positive definite form. Similarly, we arrive at a contradiction in the case r \u003d nbut to< n ... So, if the form is positive definite, then its normal form y 1 2 + y 2 2 + … + at n 2... This means that the rank and the positive index of inertia are equal n.
X The rank and positive index of inertia of the real quadratic form are n.Prove for yourself that the form is positive definite.
We note, without proof, one more theorem on positive definite real quadratic forms.
Theorem 69 ... A real quadratic form is positive definite if and only if all major minors of its matrix are positive.
Theorem 70. The squared length of a vector in any basis of the Euclidean space is given by a positive definite quadratic form.
Evidence. Let be E n – n-dimensional Euclidean space, e = (e 1 , e 2, … , e n ) Is a basis in it and D- the Gram matrix, specifying the scalar product of vectors in this basis. If a and = x 1 e 1 + x 2 e 2 + … + x n e n , in = at 1 e 1 + at 2 e 2 + … + at n e n , then ( a, in) = x T × D× atwhere x T- vector coordinates string and , y -vector coordinate column in ... Hence, and 2 = (and , and ) = x T × D× x. If we compare it with formula (60), then we get that x T × D× x is a quadratic form with a matrix G. In space E n is an orthonormal basis. In this basis and 2 = x 1 2 + x 2 2+…+ x n 2. But this means that in passing to an orthonormal basis, the quadratic form x T × D× xnormalized x 1 2 + x 2 2+…+ x n 2. By Theorem 68, we find that the form x T × D× x is positive definite.
Example. Which of the following quadratic forms are positive definite?
1. 4x 1 2 – x 1 x 2 + 3x 2 2 – x 2 x 3+ 6x 2 x 4.
2. 4x 1 x 2 – x 1 x 3 + 2x 2 2 – 4x 2 x 3+ 3x 2 x 4+ 5x 4 2 .
3. 4x 1 2 – 5x 1 x 2 + 3x 2 2 – 2x 2 x 3+ x 3 2 + 4x 2 x 4 – x 4 2 .
Decision. There are two ways to answer the question: bring the form to the canonical form, or calculate the principal minors of the matrix of the given form. For the first form we will use the first method, for the second and third - the second method.
1. 4x 1 2 – x 1 x 2 + 3x 2 2 – x 2 x 3+ 6x 2 x 4 = (4x 1 2 – x 1 x 2+ ) – + 3x 2 2 – x 2 x 3+ 6x 2 x 4 =
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§ 4. The law of inertia of quadratic forms. Classification of quadratic forms
1. The law of inertia of quadratic forms. We have already noted (see Remark 2 in item 1 of the previous section) that the rank of a quadratic form is equal to the number of nonzero canonical coefficients. Thus, the number of nonzero canonical coefficients does not depend on the choice of a nondegenerate transformation, with the help of which the form A (x, x) is reduced to the canonical form. In fact, for any method of reducing the form A (x, x) to the canonical form, the number of positive and negative canonical coefficients does not change. This property is called the law of inertia of quadratic forms.
Before proceeding to the substantiation of the law of inertia, let us make some comments.
Let the form A (x, x) in the basis e \u003d (e 1, e 2, ..., e n) be determined by the matrix A (e) \u003d (a ij):
where ξ 1, ξ 2, ..., ξ n are the coordinates of the vector x in the basis e. Suppose that this form is reduced to the canonical form using a nondegenerate transformation of coordinates
where λ 1, λ 2, ..., λ k - nonzero canonical coefficients, numbered so that the first q of these coefficients are positive, and the following coefficients are negative:
λ 1\u003e 0, λ 2\u003e 0, ..., λ q\u003e 0, λ q + 1< 0, ..., λ k <0.
Consider the following non-degenerate transformation of coordinates μ i (it is easy to see that the determinant of this transformation is nonzero):
As a result of this transformation, the form A (x, x) takes the form
called the normal kind of quadratic form.
So, with the help of some non-degenerate transformation of coordinates ξ 1, ξ 2, ..., ξ n of the vector x in the basis е \u003d (е 1, е 2, ..., е n)
(this transformation is the product of transformations ξ to μ and μ to η by formulas (7.30)) the quadratic form can be reduced to the normal form (7.31).
Let us prove the following statement.
Theorem 7.5 (law of inertia for quadratic forms). The number of terms with positive (negative) coefficients in the normal form of a quadratic form does not depend on the way the form is reduced to this form.
Evidence. Let the form A (x, x) be reduced to the normal form (7.31) with the help of a non-degenerate coordinate transformation (7.32) and, with the help of another non-degenerate coordinate transformation, reduced to the normal form
Obviously, to prove the theorem, it is sufficient to verify that the equality p \u003d q is true.
Let p\u003e q. Let us verify that in this case there is a nonzero vector x such that with respect to the bases in which the form A (x, x) has the form (7.31) and (7.33), the coordinates η 1, η 2, ..., η q and ζ p + 1, ..., ζ n of this vector are equal to zero:
η 1 \u003d 0, η 2 \u003d 0, ..., η q \u003d 0, ζ p + 1 \u003d 0, ..., ζ n \u003d 0 (7.34)
Since the coordinates η i obtained by non-degenerate transformation (7.32) of the coordinates ξ 1, ..., ξ n, and the coordinates ζ i - using a similar non-degenerate transformation of the same coordinates ξ 1, ..., ξ n, then relations (7.34) can be regarded as a system of linear homogeneous equations with respect to the coordinates ξ 1, ..., ξ n of the required vector x in the basis е \u003d ( е 1, е 2, ..., е n) (for example, in expanded form, the relation η 1 \u003d 0 has, according to (7.32), the form а 11 ξ 1 + a 12 ξ 2 + a 1 n ξ n \u003d 0) - Since p\u003e q, the number of homogeneous equations (7.34) is less than n, and therefore system (7.34) has a nonzero solution with respect to the coordinates ξ 1, ..., ξ n of the desired vector x. Therefore, if p\u003e q, then there exists a nonzero vector x for which relations (7.34) are satisfied.
Let's calculate the value of the form A (x, x) for this vector x. Turning to relations (7.31) and (7.33), we obtain
The last equality can take place only in the case η q + 1 \u003d ... \u003d η k \u003d 0 and ζ 1 \u003d ζ 2 \u003d ... \u003d ζ р \u003d 0.
Thus, in some basis, all coordinates ζ 1, ζ 2, ..., ζ n nonzero vector x are equal to zero (see the last equalities and relations (7.34)), that is, vector x is zero. Hence, the assumption p\u003e q leads to a contradiction. For similar reasons, leads to a contradiction and the assumption p< q.
So p \u003d q. The theorem is proved.
2. Classification of quadratic forms. In subsection 1 of §2 of this chapter (see Definition 2), the concepts of positive definite, negative definite, alternating sign, and quasi-definite quadratic forms were introduced.
In this subsection, using the concepts of the index of inertia, positive and negative indices of inertia of a square of a form, we will indicate how we can find out whether a quadratic form belongs to one or another of the types listed above. In this case, the index of inertia of a quadratic form is the number of nonzero canonical coefficients of this form (i.e., its rank), the positive index of inertia is the number of positive canonical coefficients, and the negative index of inertia is the number of negative canonical coefficients. It is clear that the sum of the positive and negative indices of inertia is equal to the inertia index.
So, let the index of inertia, positive and negative indices of inertia of the quadratic form A (x, x) be k, p and q (k \u003d p + q), respectively. In the previous subsection, it was proved that in any canonical basis f \u003d (f 1 , f 2, ..., fn) this form can be reduced to the following normal form:
where η 1, η 2, ..., η n are the coordinates of the vector x in the basis f.
1 °. Necessary and sufficient condition for the definite sign of a quadratic form. The following statement is true.
For the quadratic form A (x, x), given in an n -dimensional linear space L, to be definite, it is necessary and sufficient that either the positive index of inertia p or the negative index of inertia q be equal to the dimension n of the space L.
Moreover, if p \u003d n, then the form is positive definite, but if q \u003d n, then the form is negative definite.
Evidence. Since the cases of positive definite form and negative definite form are considered in a similar way, we will prove the statement for positive definite forms.
1) Necessity. Let the form A (x, x) be positive definite. Then expression (7.35) takes the form
A (x, x) \u003d η 1 2 + η 2 2 + ... + η p 2.
If at the same time p< n , то из последнего выражения следует, что для ненулевого вектора х с координатами
η 1 \u003d 0, η 2 \u003d 0, ..., η p \u003d 0, η p + 1 ≠ 0, ..., η n ≠ 0
the form A (x, x) vanishes, and this contradicts the definition of a positive definite quadratic form. Therefore, p \u003d n.
2) Sufficiency. Let p \u003d n. Then the relation (7.35) has the form А (х, х) \u003d η 1 2 + η 2 2 + ... + η р 2. It is clear that A (x, x) ≥ 0, and if A \u003d 0, then η 1 \u003d η 2 \u003d ... \u003d η n \u003d 0, that is, the vector x is zero. Therefore, A (x, x) is a positive definite form.
Comment. To clarify the question of the definiteness of a quadratic form using the indicated criterion, we must bring this form to the canonical form.
In the next subsection, we prove Sylvester's criterion for the definiteness of a quadratic form, which can be used to clarify the question of the definiteness of a form given in any basis without reduction to canonical form.
2 °. A necessary and sufficient condition for the alternation of a quadratic form. Let us prove the following statement.
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 be nonzero.
Evidence. 1) Necessity. Since the alternating form takes both positive and negative values, its representation G.35) in its normal form must contain both positive and negative terms (otherwise this form would take either non-negative or non-positive values). Therefore, both positive and negative indices of inertia are non-zero.
2) Sufficiency. Let p ≠ 0 and q ≠ 0. Then for a vector x 1, with coordinates η 1 ≠ 0, ..., η p ≠ 0, η p + 1 \u003d 0, ..., η n \u003d 0 we have A (x 1 x 1)\u003e 0, and for a vector x 2 with coordinates η 1 \u003d 0, ..., η p \u003d 0, η p + 1 ≠ 0, ..., η n ≠ 0 we have A (x 2, x 2)< 0. Следовательно, форма А(х, х) является знакопеременной.
3 °. A necessary and sufficient condition for a quadratic form to be quasi-definite. The following statement is true.
For the form A (x, x) to be quasi-definite, it is necessary and sufficient that the relations hold: either p< n
,
q = 0, либо р = 0, q < n
.
Evidence. We will consider the case of a positively quasi-sign definite form. The case of a negative quasi-sign definite form is considered similarly.
1) Necessity. Let the form A (x, x) be positively quasi-definite. Then, obviously, q \u003d 0 and p< n
(если бы р =
n
, то форма была бы положительно определенной),
2) Sufficiency. If p< n
, q = 0, то А(х, х)
≥ 0
и для ненулевого вектора х с координатами
η 1 = 0, η
2 = 0,
..., η
р = 0, η p + 1 ≠ 0, ..., η n ≠ 0 we have A (x, x) \u003d 0, i.e. A (x, x) is a positive quasi-sign definite form.
3. Sylvester's criterion (James Joseph Sylvester (1814-1897) - English mathematician) of the definite sign of a quadratic form. Let the form A (x, x) in the basis e \u003d (e 1, e 2, ..., e n) be determined by the matrix A (e) \u003d (a ij):
let it go Δ 1 \u003d a 11, - angular minors and determinant of the matrix (a ij). The following statement is true.
Theorem 7.6 (Sylvester criterion). In order for the quadratic form A (x, x) to be positive definite, it is necessary and sufficient that the inequalities Δ 1\u003e 0, Δ 2\u003e 0, ..., Δ n\u003e 0 be satisfied.
For the quadratic form to be negative definite, it is necessary and sufficient that the signs of the angle minors alternate, and Δ 1< 0.
Evidence. 1) Necessity. Let us prove first that from the condition of definiteness of the quadratic form A (x, x) it follows that Δ i ≠ 0, i \u003d 1, 2, ..., n.
Let us verify that the assumption Δ k \u003d 0 leads to a contradiction - under this assumption, there exists a nonzero vector x for which A (x, x) \u003d 0, which contradicts the definiteness of the form.
So let Δ k \u003d 0. Consider the following square homogeneous system of linear equations:
Since Δ k is the determinant of this system and Δ k \u003d 0, then the system has a nonzero solution ξ 1, ξ 2, ..., ξ k (not all ξ i are equal to 0). We multiply the first of equations (7.36) by ξ 1, the second by ξ 2, ..., the last by ξ k and add the resulting relations. As a result, we obtain the equality , the left side of which is the value of the quadratic form A (x, x) for a nonzero vector x with coordinates (ξ 1, ξ 2, ..., ξ k, 0, ..., 0). This value is equal to zero, which contradicts the definite sign of the form.
So, we made sure that Δ i ≠ 0, i \u003d 1, 2, ..., n. Therefore, we can apply the Jacobi method of reducing the form A (x, x) to the sum of squares (see Theorem 7.4) and use formulas (7.27) for the canonical coefficients λ i... If A (x, x) is a positive definite form, then all canonical coefficients are positive. But then it follows from relations (7.27) that Δ 1\u003e 0, Δ 2\u003e 0, ..., Δ n\u003e 0. If A (x, x) is a negative definite form, then all the canonical coefficients are negative. But then it follows from formulas (7.27) that the signs of the angular minors alternate, and Δ 1< 0.
2) Sufficiency. Let the conditions imposed on the angular minors Δ i in the statement of the theorem. Since Δ i ≠ 0, i \u003d 1, 2, ..., n, then the form A can be reduced to a sum of squares by the Jacobi method (see Theorem 7.4), and the canonical coefficients λ i can be found by formulas (7.27). If Δ 1\u003e 0, Δ 2\u003e 0, ..., Δ n\u003e 0, then it follows from relations (7.27) that all λ i \u003e 0, that is, the form A (x, x) is positive definite. If the signs of Δ i alternate and Δ 1< 0, то из соотношений (7.27) следует,
что форма А(х, х) отрицательно определенная. Теорема доказана.
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 among the nonzero elements there are positive and negative, and. Changing, if necessary, the numbering of the basis vectors, you can always ensure that the first elements in the diagonal matrix of the quadratic form are positive, the rest 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) takes a diagonal form, in which the coefficients of the squares of the variables are unity, minus ones, or zeros:
where the matrix of quadratic form (10.19) has the diagonal form
Definition 10.9. The notation (10.19) is called normal viewquadratic 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 arranged so that first ones come first, then minus ones, then zeros (cases of vanishing specified values,,).
Thus, the following theorem has been proved.
Theorem 10.3. Any quadratic form can be reduced to normal form (10.19) with a diagonal matrix (10.20).
Quadratic law of inertia
The quadratic form can be reduced to canonical form in various ways (by the Lagrange method, by the method of orthogonal transformations, or by the 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 invariant of the quadratic form is the rank of the quadratic form.
Theorem 10.4 (on the invariance of the rank 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 nonzero coefficients in any of its canonical forms. In other words, the rank of the quadratic form is equal to the number of nonzero eigenvalues \u200b\u200bof 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 the normal form (3) of the quadratic form are called positive and negative indices inertia of the quadratic form, respectively. The list is called by signature quadratic form.
Positive and negative indices of inertia are numerical invariants of the quadratic form. The following theorem is valid called law of inertia.
Theorem 10.5 (law of inertia ) The canonical form (10.17) of the quadratic form is uniquely determined, 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 ():
it is mandatory, that is, the number of positive coefficients coincides with the number of positive coefficients.
Contrary to the statement, suppose that. Since transformations (10.21) are non-degenerate, we can express the canonical variables from them:
Let us find a vector such that the corresponding vectors have the form
For this, we represent the matrices in the following block forms:
where the -matrix, -matrix, -matrix, -matrix are denoted.
As a result of the block representations of the matrices and we 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 components). Since, 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 nonzero solution can be distinguished.
On the resulting vector, the shape values \u200b\u200bhave different signs:
which is impossible. Hence, the assumption about what is wrong, that is.
From what 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 in three variables:
two nonsingular linear transformations, with the corresponding matrices
(the first matrix corresponds to the Lagrange method, the second to the orthogonal transformation method) is reduced, respectively, to two different canonical forms
Moreover, both canonical forms have the same signature
6. Sign-definite and sign-alternating quadratic forms
Quadratic forms are subdivided into types depending on the set of values \u200b\u200bthey accept.
Definition 10.11.The quadratic form is called:
positively defined
negatively definedif for any nonzero vector:;
nonpositively definite (negatively semidefinite)if for any nonzero vector:;
non-negative definite (positive semidefinite)if for any nonzero vector:;
alternatingif there are nonzero vectors,:.
Definition 10.12. Positive (negative) definite quadratic forms are called definite... Nonpositively (nonnegatively) definite quadratic forms are called permanent.
The type of a quadratic form can be easily determined by converting it to the canonical (or normal) form. The following two theorems are true.
Theorem 10.6. Let the quadratic form be reduced to the canonical form and have the signature (,). Then:
Is an positively defined ;
Is an negatively defined ;
Is an not positively definite ;
Is an non-negative definite ;
Is an alternating .). Then: non-negative definite for all;
Is an alternating among the eigenvalues \u200b\u200bthere are both positive and negative.
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