Positive definite quadratic forms

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

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

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

returned to normal

.

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

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

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

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

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

, , , … , .

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

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

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

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

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

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

Square shapes.
Significance of forms. Sylvester's criterion

The adjective "square" immediately suggests that something here is connected with a square (second degree), and very soon we will know this "something" and what a form is. Turned out straight away :)

Welcome to my new lesson, and as an immediate warm-up, we will look at the striped shape linear. Linear form variables called homogeneous 1st degree polynomial:

- some specific numbers * (we assume that at least one of them is different from zero), and are variables that can take arbitrary values.

* In this topic, we will only consider real numbers .

We have already encountered the term "homogeneous" in the lesson about homogeneous systems of linear equations, and in this case it implies that the polynomial does not have an added constant .

For example: – linear form of two variables

Now the shape is quadratic. quadratic form variables called homogeneous 2nd degree polynomial, each term of which contains either the square of the variable or double product of variables. So, for example, the quadratic form of two variables has the following form:

Attention! This is a standard entry, and you do not need to change anything in it! Despite the “terrible” look, everything is simple here - double subscripts of constants signal which variables are included in one or another term:
– this term contains the product and (square);
- here is the work;
- and here is the work.

- I immediately anticipate a gross mistake when they lose the "minus" of the coefficient, not realizing that it refers to the term:

Sometimes there is a "school" version of the design in the spirit, but then only sometimes. By the way, note that the constants here do not tell us anything at all, and therefore it is more difficult to remember the “easy notation”. Especially when there are more variables.

And the quadratic form of three variables already contains six terms:

... why are "two" multipliers put in the "mixed" terms? This is convenient, and it will soon become clear why.

However, we will write down the general formula, it is convenient to arrange it with a “sheet”:

- carefully study each line - there's nothing wrong with that!

The quadratic form contains terms with squared variables and terms with their pair products (cm. combinatorial formula of combinations) . Nothing else - no "lonely x" and no added constant (then you get not a quadratic form, but heterogeneous 2nd degree polynomial).

Matrix notation of a quadratic form

Depending on the values, the considered form can take both positive and negative values, and the same applies to any linear form - if at least one of its coefficients is non-zero, then it can turn out to be either positive or negative (depending on values).

This form is called alternating. And if everything is transparent with the linear form, then things are much more interesting with the quadratic form:

It is quite clear that this form can take on the values ​​of any sign, thus, the quadratic form can also be alternating.

It may not be:

– always, unless both are equal to zero.

- for anyone vector except for zero.

And generally speaking, if for any non-zero vector , , then the quadratic form is called positive definite; if - then negative definite.

And everything would be fine, but the definiteness of the quadratic form is visible only in simple examples, and this visibility is lost already with a slight complication:
– ?

One might assume that the form is positively defined, but is it really so? Suddenly there are values ​​at which it is less than zero?

On this account, there theorem: if all eigenvalues matrices of quadratic form are positive * , then it is positively defined. If all are negative, then it is negative.

* It is proved in theory that all eigenvalues ​​of a real symmetric matrix valid

Let's write the matrix of the above form:
and from the equation let's find her eigenvalues:

We solve the good old quadratic equation:

, so the form is positively defined, i.e. for any non-zero values, it is greater than zero.

The considered method seems to be working, but there is one big BUT. Already for the “three by three” matrix, looking for eigenvalues ​​is a long and unpleasant task; with a high probability you get a polynomial of the 3rd degree with irrational roots.

How to be? There is an easier way!

Sylvester's criterion

No, not Sylvester Stallone :) First, let me remind you what angular minors matrices. This is determinants which "grow" from its upper left corner:

and the last one is exactly equal to the determinant of the matrix.

Now, in fact, criterion:

1) Quadratic form defined positively if and only if ALL of its angular minors are greater than zero: .

2) Quadratic form defined negative if and only if its angular minors alternate in sign, while the 1st minor is less than zero: , , if is even or , if is odd.

If at least one angular minor has the opposite sign, then the form sign-alternating. If the angular minors are of “that” sign, but there are zeros among them, then this is a special case, which I will analyze a little later, after we have gone over the more common examples.

Let us analyze the angular minors of the matrix :

And this immediately tells us that the form is not negatively determined.

Conclusion: all angle minors are greater than zero, so the shape positively defined.

Is there a difference with the eigenvalue method? ;)

We write the shape matrix from Example 1:

its first angular minor, and the second , whence it follows that the form is sign-alternating, i.e. depending on the values ​​, can take both positive and negative values. However, this is so obvious.

Take the form and its matrix from Example 2:

here at all without insight not to understand. But with the Sylvester criterion, we don’t care:
, hence the form is definitely not negative.

, and definitely not positive. (because all angle minors must be positive).

Conclusion: the shape is alternating.

Warm-up examples for self-solving:

Example 4

Investigate quadratic forms for sign-definiteness

a)

In these examples, everything is smooth (see the end of the lesson), but in fact, to complete such a task Sylvester's criterion may not be sufficient.

The point is that there are "boundary" cases, namely: if for any non-zero vector , then the shape is defined non-negative, if - then non-positive. These forms have non-zero vectors for which .

Here you can bring such a "button accordion":

Highlighting full square, we immediately see non-negativity form: , moreover, it is equal to zero for any vector with equal coordinates, for example: .

"Mirror" example non-positive certain form:

and an even more trivial example:
– here the form is equal to zero for any vector , where is an arbitrary number.

How to reveal the non-negativity or non-positiveness of a form?

For this we need the concept major minors matrices. The main minor is a minor composed of elements that are at the intersection of rows and columns with the same numbers. So, the matrix has two principal minors of the 1st order:
(the element is at the intersection of the 1st row and 1st column);
(the element is at the intersection of the 2nd row and 2nd column),

and one major 2nd order minor:
- composed of elements of the 1st, 2nd row and 1st, 2nd column.

Matrix "three by three" There are seven main minors, and here you already have to wave your biceps:
- three minors of the 1st order,
three minors of the 2nd order:
- composed of elements of the 1st, 2nd row and 1st, 2nd column;
- composed of elements of the 1st, 3rd row and 1st, 3rd column;
- composed of elements of the 2nd, 3rd row and 2nd, 3rd column,
and one 3rd order minor:
- composed of elements of the 1st, 2nd, 3rd row and 1st, 2nd and 3rd columns.
Exercise for understanding: write down all the main minors of the matrix .
We check at the end of the lesson and continue.

Schwarzenegger criterion:

1) Non-zero* quadratic form defined non-negative if and only if ALL of its principal minors non-negative(greater than or equal to zero).

* The zero (degenerate) quadratic form has all coefficients equal to zero.

2) Nonzero quadratic form with matrix defined non-positive if and only if its:
– principal minors of the 1st order non-positive(less than or equal to zero);
are principal minors of the 2nd order non-negative;
– principal minors of the 3rd order non-positive(alternation has begun);
…
– major minor of the th order non-positive, if is odd or non-negative, if is even.

If at least one minor is of the opposite sign, then the form is sign-alternating.

Let's see how the criterion works in the above examples:

Let's make a shape matrix, and primarily let's calculate the angular minors - what if it is positively or negatively defined?

The obtained values ​​do not satisfy the Sylvester criterion, however, the second minor not negative, and this makes it necessary to check the 2nd criterion (in the case of the 2nd criterion, it will not be fulfilled automatically, i.e., a conclusion is immediately made about the sign alternation of the form).

Major minors of the 1st order:
- are positive
2nd order major minor:
- not negative.

Thus, ALL major minors are non-negative, so the form non-negative.

Let's write the form matrix , for which, obviously, the Sylvester criterion is not satisfied. But we also did not receive opposite signs (because both angular minors are equal to zero). Therefore, we check the fulfillment of the criterion of non-negativity / non-positiveness. Major minors of the 1st order:
- not positive
2nd order major minor:
- not negative.

Thus, according to the Schwarzenegger criterion (point 2), the form is determined non-positively.

Now, fully armed, we will analyze a more entertaining problem:

Example 5

Examine the quadratic form for sign-definiteness

This form is decorated with the order "alpha", which can be equal to any real number. But it'll only be more fun decide.

First, let's write down the form matrix, probably, many have already adapted to do it orally: on main diagonal we put the coefficients at the squares, and at the symmetrical places - the half coefficients of the corresponding "mixed" products:

Let's calculate the angular minors:

I will expand the third determinant along the 3rd line:

A quadratic form is a homogeneous polynomial of the 2nd degree in several variables.

The quadratic form in variables consists of terms of two types: the squares of the variables and their pairwise products with some coefficients. It is customary to write the quadratic form in the form of the following square scheme:

Pairs of similar terms are written with the same coefficients, so that each of them is half the coefficient of the corresponding product of the variables. Thus, each quadratic form is naturally associated with its coefficient matrix, which is symmetric.

It is also convenient to represent the quadratic form in the following matrix notation. Denote by X a column of variables by X - a row, i.e., a matrix transposed with X. Then

Quadratic forms are found in many branches of mathematics and its applications.

In number theory and crystallography, quadratic forms are considered under the assumption that the variables take only integer values. In analytic geometry, the quadratic form is part of the equation of a curve (or surface) of order. In mechanics and physics, the quadratic form appears to express the kinetic energy of the system in terms of the components of generalized velocities, etc. But, in addition, the study of quadratic forms is also necessary in analysis when studying functions of many variables, in questions for the solution of which it is important to find out how the given function in the vicinity of the given point deviates from the linear function approximating it. An example of a problem of this type is the study of a function for maximum and minimum.

Consider, for example, the problem of exploring the maximum and minimum for a function of two variables that has continuous partial derivatives up to order. A necessary condition for a point to give a maximum or minimum of a function is the equality to zero of the partial derivatives of the order at the point. Let's assume that this condition is met. We give the variables x and y small increments and k and consider the corresponding increment of the function. According to the Taylor formula, this increment, up to small higher orders, is equal to the quadratic form where are the values ​​of the second derivatives calculated at the point If this quadratic form is positive for all values ​​of and k (except then the function has a minimum at a point; if it is negative, then it has a maximum. Finally, if the shape takes on both positive and negative values, then there will be no maximum or minimum. Functions of a larger number of variables are studied in a similar way.

The study of quadratic forms mainly consists in the study of the problem of the equivalence of forms with respect to one or another set of linear transformations of variables. Two quadratic forms are said to be equivalent if one of them can be translated into the other by means of one of the transformations of the given set. Closely related to the problem of equivalence is the problem of reduction of form, i.e. converting it to some possibly simplest form.

In various questions related to quadratic forms, various sets of admissible transformations of variables are also considered.

In questions of analysis, any non-singular transformations of variables are applied; For the purposes of analytic geometry, orthogonal transformations are of greatest interest, i.e., those that correspond to the transition from one system of variable Cartesian coordinates to another. Finally, in number theory and in crystallography, linear transformations with integer coefficients and with a determinant equal to one are considered.

We will consider two of these problems: the question of reducing a quadratic form to its simplest form by means of any non-singular transformations, and the same question for orthogonal transformations. First of all, let's find out how a matrix of a quadratic form is transformed under a linear transformation of variables.

Let , where A is a symmetric matrix of form coefficients, X is a column of variables.

Let's make a linear transformation of variables, writing it in abbreviated form . Here C denotes the matrix of coefficients of this transformation, X is a column of new variables. Then and hence, so that the matrix of the transformed quadratic form is

The matrix automatically turns out to be symmetric, which is easily verified. Thus, the problem of reducing a quadratic form to its simplest form is equivalent to the problem of reducing a symmetric matrix to its simplest form by multiplying it from the left and right by mutually transposed matrices.

Quadratic forms

quadratic form f(x 1, x 2,..., x n) of n variables is called the sum, each term of which is either the square of one of the variables, or the product of two different variables, taken with a certain coefficient: f(x 1, x 2, ...,x n) = (a ij = a ji).

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

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

Indeed

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

To do this, we find a matrix of a quadratic form. Its diagonal elements are equal to the coefficients at the squares of the variables, and the remaining elements are equal to half of the corresponding coefficients of the quadratic form. So

Let the matrix-column of variables X be obtained by a nondegenerate linear transformation of the matrix-column Y, i.e. X = CY, where C is a non-degenerate matrix of order n. Then the quadratic form
f(X) \u003d X T AX \u003d (CY) T A (CY) \u003d (Y T C T) A (CY) \u003d Y T (C T AC) Y.

Thus, under a non-degenerate linear transformation C, the matrix of the quadratic form takes the form: A * = C T AC.

For example, let's find the quadratic form f(y 1, y 2) obtained from the quadratic form f(x 1, x 2) = 2x 1 2 + 4x 1 x 2 - 3x 2 2 by a linear transformation.

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

Its matrix is ​​diagonal.

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

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

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

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

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

f (x 1, x 2, x 3) \u003d 2 (x 1 + x 2) 2 - 5 (x 2 2 - 2 * x 2 * (1/10) x 3 + (1/100) x 3 2) - (5/100) x 3 2 =
\u003d 2 (x 1 + x 2) 2 - 5 (x 2 - (1/10) x 3) 2 - (1/20) x 3 2.

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

Note that the canonical form of a quadratic form is defined ambiguously (the same quadratic form can be reduced to the canonical form in different ways). However, canonical forms obtained by various methods have a number of common properties. In particular, the number of terms with positive (negative) coefficients of a quadratic form does not depend on how the form is reduced to this form (for example, in the considered example there will always be two negative and one positive coefficient). This property is called the law of inertia of quadratic forms.

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

It should also be noted that the rank of a matrix of a quadratic form, called the rank of the quadratic form, is equal to the number of non-zero coefficients of the canonical form and does not change under linear transformations.

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

For example, the quadratic form f 1 (X) \u003d x 1 2 + x 2 2 is positive definite, because is the sum of squares, and the quadratic form f 2 (X) \u003d -x 1 2 + 2x 1 x 2 - x 2 2 is negative definite, because represents it can be represented as f 2 (X) \u003d - (x 1 - x 2) 2.

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

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

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

Major (corner) minor The k-th order of the matrix A of the n-th order is called the determinant of the matrix, composed of the first k rows and columns of the matrix A ().

Note that for negative-definite quadratic forms, the signs of the principal minors alternate, and the first-order minor must be negative.

For example, we examine the quadratic form f (x 1, x 2) = 2x 1 2 + 4x 1 x 2 + 3x 2 2 for sign-definiteness.

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

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

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

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

In this section we will focus on a special but important class of positive quadratic forms.

Definition 3. A real quadratic form is called non-negative (non-positive) if for any real values ​​of the variables

. (35)

In this case, the symmetric matrix of coefficients is called positive semidefinite (negative semidefinite).

Definition 4. A real quadratic form is called positive-definite (negative-definite) if for any real values ​​of the variables that are not simultaneously equal to zero

. (36)

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

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

Let a non-negative form be given. We represent it as a sum of independent squares:

. (37)

In this representation, all squares must be positive:

. (38)

Indeed, if there were any , then it would be possible to choose such values ​​for which

But then, for these values ​​of the variables, the form would have a negative value, which is impossible by the condition. Obviously, conversely, from (37) and (38) it follows that the form is positive.

Thus, a non-negative quadratic form is characterized by the equalities .

Let now be a positive definite form. Then also the non-negative form. Therefore, it can be represented in the form (37), where all are positive. It follows from the positive definiteness of the form that . Indeed, in the case it is possible to choose such values ​​that are not simultaneously equal to zero, for which all would vanish. But then, by virtue of (37), at , which contradicts condition (36).

It is easy to see that, conversely, if in (37) and are all positive, then is a positive definite form.

In other words, a non-negative form is positive definite if and only if it is not singular.

The following theorem gives a criterion for the positive definiteness of a form in the form of inequalities that the coefficients of the form must satisfy. In this case, the notation already encountered in the previous sections for successive principal minors of the matrix is ​​used:

.

Theorem 3. For a quadratic form to be positive definite, it is necessary and sufficient that the inequalities

Proof. The sufficiency of conditions (39) follows directly from the Jacobi formula (28). The necessity of conditions (39) is established as follows. From the positive definiteness of the form follows the positive definiteness of the "truncated" forms

.

But then all these forms must be non-singular, i.e.

Now we have the opportunity to use the Jacobi formula (28) (for ). Since on the right side of this formula all squares must be positive, then

This implies inequalities (39). The theorem has been proven.

Since any principal minor of a matrix, with proper renumbering of variables, can be placed in the upper left corner, we have

Consequence. In positive definite quadratic form, all principal minors of the coefficient matrix are positive:

Comment. From the non-negativity of successive principal minors

does not follow the non-negativity of the form . Indeed, the form

,

wherein , satisfies the conditions , but is not non-negative.

However, there is the following

Theorem 4. For a quadratic form to be non-negative, it is necessary and sufficient that all principal minors of its coefficient matrix be non-negative:

Proof. Let us introduce an auxiliary form that is nonpositive, it is necessary and sufficient that the inequalities