It was found that the number of nonzero 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 the quadratic form does not depend on the method of reducing the quadratic form to the normal form.
Let the quadratic form f rank r from n unknown x 1 , x 2 , …, x n brought to normal form in two ways, that is
f = + 
+ … + 
– 
– … – 
,
f = 
+ 
+ … + 
– 
– … – 
... It can be proved that k = l.
Definition 11.14. The number of positive squares in normal form to which a real quadratic form is reduced is called positive index of inertia this form; number of negative squares - negative index of inertia, and their sum is inertia index quadratic or signature shape f.
If p- positive index of inertia; q- negative index of inertia; k = r = p + q Is the inertia index.
Classification of quadratic forms
Let the quadratic form A(x, x) the index of inertia is k, the positive index of inertia is p, the negative index of inertia is q, then k = p + q.
It was proved 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 the basis ( f}.
A necessary and sufficient condition for the definite sign of a quadratic form
Statement11.1. A(x, x) given in n V, was definite, it is necessary and sufficient that either a positive index of inertia p, or negative index of inertia 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 negatively defined (that is, for any x ≠ 0 A(x, x) < 0).
A necessary and sufficient condition for the alternating sign of a quadratic form
Statement 11.2. For the quadratic form A(x, x) given in n-dimensional vector space V, was alternating(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.
A necessary and sufficient condition for the quasi-sign-variability of a quadratic form
Statement 11.3. For the quadratic form A(x, x) given in n-dimensional vector space V, was quasi-variable(that is, for any vector x or A(x, x) ≥ 0 or A(x, x) ≤ 0 and there is a nonzero vector x, what A(x, x) = 0) is necessary and sufficient for one of the two relations to hold: p < n, q= 0 or p = 0, q < n.
Comment... In order to apply these features, the quadratic form must be reduced to the canonical form. This is not required in the criterion of sign-definiteness of Sylvester 15.
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 ones, 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 view 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 arranged so that first ones come first, then minus ones, then zeros (cases of vanishing specified values,,).
Thus, we have proved the following theorem.
Theorem 10.3. Any quadratic form can be reduced to normal form (10.19) with diagonal matrix (10.20).
Quadratic law of inertia
Quadratic form can be reduced to canonical form different ways(by the Lagrange method, the method of orthogonal transformations or 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. These are 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 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 the normal form (3) of the quadratic form are called positive and negative indices inertia of the quadratic form, respectively. In this case, the list is called signature quadratic form.
Positive and negative indices of inertia are numerical invariants of the quadratic form. There is a theorem 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 way the quadratic form is reduced to the canonical form).
□ The statement of the theorem means that if one and the same quadratic form using two non-singular linear transformations
reduced to various canonical forms ():
then it is obligatory, 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
To do 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 will compose a homogeneous system of linear algebraic equations, taking from (10.22) the first equations, and from (10.23) - the last equations:
The resulting system contains equations and unknowns (vector components). 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 nonzero solution can be distinguished.
On the resulting vector, the shape values have 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 non-singular 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 they accept.
Definition 10.11. The quadratic form is called:
positively defined
negatively defined if for any nonzero vector:;
nonpositively definite (negatively semidefinite) if for any nonzero vector:;
nonnegatively definite (positively semidefinite) if for any nonzero vector:;
alternating if 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 there are both positive and negative.
September proved 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 stocks. Popular in last years bond funds, as well as Sberbank shares, which could be hit hardest in the event of tougher US sanctions.
Vitaly Kapitonov
Five months later, gold was the most profitable investment of the month. According to "Money", having invested on August 15 in a precious metal 100 thousand rubles, 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 inflationary expectations in the US. 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 strengthened by the growth of the dollar in Russia (+ 2.5%). As a result, ruble investments in the precious metal have brought significant income.
However, further investments in gold should be treated with caution, according to market participants. Key risk for investments in the precious metal, the escalation of the trade confrontation between the United States and China remains. "The factor of political pressure has been ruled out, which means that the emergence of new barriers is practically a thing of the past. Such a 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.
Among the most profitable financial products shares remain investment funds, and individual products of asset management companies were able to provide margins exceeding that of gold. In October, the most successful investments were in industry-specific equity funds focused on metallurgy, 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 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 attracted volume 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 (%)
|
Sources: National League of Governors, Investfunds.
August outsiders - shares - climbed 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 15 to 18 August, the MICEX index fell by 1.2%. However, the situation improved after 24 August. In three weeks, the index jumped almost 5% and climbed 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, 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: an investor could earn 4.2-8.3 thousand rubles on these securities. 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 direct 75% net profit for the payment of dividends.
An exception to the general picture was the shares of RusHydro, Rostelecom, Aeroflot, investments in which would have brought a loss in the amount of 200 rubles. up to 1.4 thousand rubles. The maximum losses would be for investors who have invested in securities Sberbank - 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. "The rebound in the securities of the largest Russian bank is very likely, and the risks of their purchases are fully justified. For the time being, medium-term investors should be guided by profit-taking in the region of 180 rubles. per share ", - said analyst" ALOR Broker "Alexei Antonov.
What income did investments in stocks bring (%)
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Over 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 corner minors of its matrix are strictly positive.
- A quadratic form is negative definite if and only if the signs of all corner minors of its matrix alternate, and the minor of order 1 is negative.
The bilinear form, polar to the positive definite quadratic form, satisfies all the axioms of the dot product.
Canonical view
Real case
In the case when 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 + ⋯ + xp 2 - xp + 1 2 - ⋯ - xp + 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) is the rank of the quadratic form. In the case of a nondegenerate quadratic form p + q = n (\ displaystyle p + q = n), and in the case of a degenerate one - p + q<
n
{\displaystyle p+q
To reduce the quadratic form to the canonical form, the Lagrange method or orthogonal transformations of the basis are usually used, and this quadratic form can be reduced to the canonical form not in one, but in many ways.
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 called a pair (p, q) (\ displaystyle (p, q))... Numbers p, q, p - q (\ displaystyle p, q, p-q) are invariants of the quadratic form, i.e. do not depend on the method of its reduction to the canonical form ( Sylvester's law of inertia).
Complex case
In the case when K = C (\ displaystyle K = \ mathbb (C))(the 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 + ⋯ + xr 2, (∗ ∗) (\ displaystyle Q (x) = x_ (1) ^ (2) + \ cdots + x_ (r) ^ (2), \ qquad ( **))where r (\ displaystyle r) is the rank of the quadratic form. Thus, in the complex case (as opposed to the real one), the quadratic form has one single invariant - rank, and all non-degenerate forms have the same canonical form (sum of squares).
