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 nondegenerate 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.
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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 non-degenerate 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 = (e 1, e 2, ..., e n) be determined by the matrix A (e) = (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> 0, λ 2> 0, ..., λ q> 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) will take the form
called the normal kind of quadratic form.
So, with the help of some nondegenerate transformation of coordinates ξ 1, ξ 2, ..., ξ n of the vector x in the basis е = (е 1, е 2, ..., е n)
(this transformation is the product of the 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 the quadratic form does not depend on the way the form is reduced to this form.
Proof. 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 suffices to verify the equality p = q.
Let p> 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 = 0, η 2 = 0, ..., η q = 0, ζ p + 1 = 0, ..., ζ n = 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 considered as a system of linear homogeneous equations with respect to the coordinates ξ 1, ..., ξ n of the required vector x in the basis e = ( e 1, e 2, ..., e n) (for example, in expanded form, the relation η 1 = 0 has, according to (7.32), the form a 11 ξ 1 + a 12 ξ 2 + a 1 n ξ n= 0) - Since p> 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> 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 = ... = η k = 0 and ζ 1 = ζ 2 = ... = ζ p = 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. Consequently, the assumption p> q leads to a contradiction. For similar reasons, leads to a contradiction and the assumption p< q.
So p = 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 shape, 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 inertia index be positive and negative indices the inertia of the quadratic form A (x, x) are, respectively, equal to k, p and q (k = p + q). In the previous section it was proved that in any canonical basis f = (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 °. A 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 = n, then the form is positive definite, but if q = n, then the form is negative definite.
Proof. Since the cases of positive definite form and negative definite form are considered in a similar way, we will carry out the proof of 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) = 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 = 0, η 2 = 0, ..., η p = 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 = n.
2) Sufficiency. Let p = n. Then relation (7.35) has the form A (x, x) = η 1 2 + η 2 2 + ... + η р 2. It is clear that A (x, x) ≥ 0, and if A = 0, then η 1 = η 2 = ... = η n= 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 with the help of 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 the canonical form.
2 °. A necessary and sufficient condition for the alternation of a quadratic form. Let us prove the following statement.
In order for the quadratic form to be alternating, it is necessary and sufficient that both the positive and negative indices of inertia of this form be nonzero.
Proof. 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 = 0, ..., η n = 0 we have A (x 1 x 1)> 0, and for a vector x 2 with coordinates η 1 = 0, ..., η p = 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
.
Proof. We will consider the case of a positively quasi-sign definite form. The case of a negative quasi-sign definite form is considered in a similar way.
1) Necessity. Let the form A (x, x) be positively quasi-definite. Then, obviously, q = 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) = 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 = (e 1, e 2, ..., e n) be determined by the matrix A (e) = (a ij):
let it go Δ 1 = a 11, 
- angular minors and the determinant of the matrix (a ij). The following statement is true.
Theorem 7.6 (Sylvester criterion). For the quadratic form A (x, x) to be positive definite, it is necessary and sufficient that the inequalities Δ 1> 0, Δ 2> 0, ..., Δ n> 0 be satisfied.
For the quadratic form to be negative definite, it is necessary and sufficient that the signs of the angular minors alternate, and Δ 1< 0.
Proof. 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 = 1, 2, ..., n.
Let us verify that the assumption Δ k= 0 leads to a contradiction - under this assumption there exists a nonzero vector x for which A (x, x) = 0, which contradicts the definite sign of the form.
So let Δ k= 0. Consider the following square homogeneous system of linear equations:
Since Δ k is the determinant of this system and Δ k= 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 obtained 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 = 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> 0, Δ 2> 0, ..., Δ n> 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 = 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> 0, Δ 2> 0, ..., Δ n> 0, then it follows from relations (7.27) that all λ i> 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.
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 only one invariant - rank, and all non-degenerate forms have the same canonical form (sum of squares).
So, according to the theorem on reduction of a quadratic form, for any quadratic form \ (A (x, x) \) there exists a canonical basis \ (\ (f_1, \, f_2, ..., f_n \) \), so that for any vectors \ (x \), \ [x = \ sum _ (k = 1) ^ n \ eta _kf_k, \ quad A (x, x) = \ sum _ (k = 1) ^ n \ lambda _k \ eta _k ^ 2. \] Since \ (A (x, x) \) is real-valued, and our base changes also include only real numbers, we conclude that the numbers \ (\ lambda _k \) are real. These numbers include positive, negative and zero.
Definition. The number \ (n _ + \) positive numbers \ (\ lambda _k \) is called positive index of the quadratic form \ (A (x, x) \), the number \ (n _- \) negative numbers \ (\ lambda _k \) is called negative subscript of the quadratic form , the number \ ((n _ ++ n _-) \) is called the rank of the quadratic form ... If \ (n _ + = n \), the quadratic form is called positive .
Generally speaking, the reduction of a quadratic form to a diagonal form is not realized in a unique way. The question arises: do the numbers \ (n _ + \), \ (n _- \) depend on the choice of a basis in which the quadratic form is diagonal?
Theorem (The law of inertia of quadratic forms). The positive and negative indices of a quadratic form do not depend on the way it is reduced to the canonical form.
Let there be two canonical bases, \ (\ (f \) \), \ (\ (g \) \), so that any vector \ (x \) can be represented as: \ [x = \ sum_ (k = 1) ^ n \ eta _kf_k = \ sum _ (m = 1) ^ n \ zeta _mg_m, \] and \ [A (x, x) = \ sum_ (k = 1) ^ n \ lambda _k \ eta _k ^ 2 = \ sum _ (m = 1) ^ n \ mu _m \ zeta _m ^ 2. \ quad \ quad (71) \] Suppose that among \ (\ lambda _k \) the first \ (p \) are positive, the rest are either negative or zeros, among \ (\ mu_m \) the first \ (s \) are positive, the rest are either negative or zero. We need to prove that \ (p = s \). We rewrite (71): \ [\ sum_ (k = 1) ^ p \ lambda _k \ eta _k ^ 2- \ sum _ (m = s + 1) ^ n \ mu _m \ zeta _m ^ 2 = - \ sum_ ( k = p + 1) ^ n \ lambda _k \ eta _k ^ 2 + \ sum _ (m = 1) ^ s \ mu _m \ zeta _m ^ 2, \ quad \ quad (72) \] so that all terms in both sides of the equality are non-negative. Suppose \ (p \) and \ (s \) are not equal, for example \ (p
We have proven that the positive indices are the same. Similarly, one can prove that negative indices are the same. h.t.d.
1. Convert quadratic forms to the sum of squares:
a) \ (x_1 ^ 2 + 2x_1x_2 + 2x_2 ^ 2 + 4x_2x_3 + 5x_3 ^ 2 \);