Analysis, matrix. Matrix Analysis Matrix Strategy Development Method

11.3. Matrix method for developing strategies Development of vision Organizations of the external and internal environment of organizations explain the diversity of organizations themselves and their actual state. The parameters determining the position of each

Course of lectures on discipline

"Matrix Analysis"

for students II students

mathematical faculty specialty

"Economic cybernetics"

(lecturer Dmitruk Maria Aleksandrovna)

1. Defining a function.

DF. Let

The solution to this problem is known when F (X) is a polynomial:

Definition F (A) in the general case.

Let M (x) be the minimum polynomial A and it has such a canonical decomposition

Let G (a) \u003d h (a) (1) be (1), then the polynomial D (x) \u003d g (x) -h (x) is the annuling polynomial for A, since D (a) \u003d 0, therefore, D (x ) It is divided into a linear polynomial, i.e. d (x) \u003d m (x) * Q (x) (2).

Consistent M numbers for F (X) such

If the set F (SP A) is defined for f (x), the function is determined on the spectrum of the matrix A.

From (3) it follows that the polynomials h (x) and g (x) have the same values \u200b\u200bon the spectrum of the matrix A.

Our reasoning is reversible, i.e. From (3) þ (3) þ (1). Thus, if the matrix A is specified, then the value of the polynomial F (x) is determined by the values \u200b\u200bof this polynomial on the spectrum of the matrix A, i.e. All polynomials G i (x), receiving the same values \u200b\u200bon the matrix spectrum have the same matrix values \u200b\u200bG i (a). We will require that the definition of the value F (a) in the general case submitted to the same principle.

The values \u200b\u200bof the function f (x) on the spectrum of the matrix A should be fully determined f (a), i.e. The functions having the same values \u200b\u200bon the spectrum should have the same matrix value F (a). It is obvious that to determine F (a), in general, it is sufficient to find a polynomial G (x), which would take the same values \u200b\u200bon the spectrum A, as well as the function f (a) \u003d G (a).

DF. If f (x) is defined on the spectrum of the matrix A, then f (a) \u003d G (a), where G (a) is a polynomial that takes the same meanings on the spectrum as f (a),

DF.The value of the function from the matrix A Let's call the value of the polynomial from this matrix when

Among the polynomials from [x], taking the same values \u200b\u200bon the matrix spectrum, as, and F (x), the degree is not higher (M-1), which takes the same values \u200b\u200bon the spectrum A, and F (X) is the balance of division Any polynomial G (x) having the same values \u200b\u200bon the matrix spectrum, as well as f (x), to the minimum polynomial M (x) \u003d G (x) \u003d m (x) * G (x) + R (x) .

This polynomial R (x) is called the interpolation polynomial of Lagrange Sylvester for the function f (x) on the spectrum of the matrix A.

Comment. If the minimum polynomial M (x) matrix is \u200b\u200bnot multiple roots, i.e.

Example:

Find R (x) for arbitrary f (x) if the matrix

. We construct F (H 1). We find the minimum polynomial H 1 - the last invariant multiplier:

, D n - 1 \u003d x 2; D n - 1 \u003d 1;

m x \u003d f n (x) \u003d d n (x) / d n - 1 (x) \u003d x nÞ 0 - N -Conal root M (x), i.e. N-multiple eigenvalues \u200b\u200bH 1.

, r (0) \u003d f (0), r '(0) \u003d f' (0), ..., r (n-1) (0) \u003d f (n - 1) (0)Þ .

2. Properties of functions from matrices.

Property number 1. If the matrix

Evidence:

Let the characteristic polynomial of the matrix A takes:

We will replace in equality:

Equality (*) is valid for any set F (x), so replace a polynomial F (X) on

On the left, we received a characteristic polynomial for the matrix F (a), laid out on the right to linear multipliers, from where it follows that

Scold.

Property number 2. Let the matrix

Evidence:

Because The function f (x) is defined on the spectrum of the matrix A, then there is an interpolation polynomial matrix R (x) such that

Course of lectures on discipline

"Matrix Analysis"

for students II students

mathematical faculty specialty

"Economic cybernetics"

(lecturer Dmitruk Maria Aleksandrovna)

Chapter 3. Functions from matrices.

1. Defining a function.

DF. Let the function of the scalar argument. It is required to determine what to understand under F (a), i.e. It is necessary to extend the function f (x) to the matrix value of the argument.

The solution of this problem is known when f (x) polynomial:, then.

Definition F (A) in the general case.

Let M (x) be the minimum polynomial A and it has such a canonical decomposition, eigenvalues \u200b\u200bA. Let the polynomials G (x) and h (x) take the same values.

Let G (a) \u003d h (a) (1), then the polynomial D (x) \u003d g (x) -h (x) \u003d G (x) -h (x), the annuling polynomial for A, since D (a) \u003d 0, therefore, D (x) It is divided into linear polynomial, i.e. d (x) \u003d m (x) * Q (x) (2).

Then, i.e. (3) ,.

We agree M numbers for f (x) such to call the values \u200b\u200bof the function f (x) on the spectrum of the matrix A, and the set of these values \u200b\u200bwill be denoted.

If the set F (SP A) is defined for f (x), the function is determined on the spectrum of the matrix A.

From (3) it follows that the polynomials h (x) and g (x) have the same values \u200b\u200bon the spectrum of the matrix A.

Our reasoning is reversible, i.e. From (3) (3) (1). Thus, if the matrix A is specified, then the value of the polynomial F (x) is determined by the values \u200b\u200bof this polynomial on the spectrum of the matrix A, i.e. All polynomials Gi (x) accepting the same values \u200b\u200bon the matrix spectrum have the same Matrix values \u200b\u200bGi (A). We will require that the definition of the value F (a) in the general case submitted to the same principle.

The values \u200b\u200bof the function f (x) on the spectrum of the matrix A should be fully determined f (a), i.e. The functions having the same values \u200b\u200bon the spectrum should have the same matrix value F (a). It is obvious that to determine F (a), in general, it is sufficient to find a polynomial G (x), which would take the same values \u200b\u200bon the spectrum A, as well as the function f (a) \u003d G (a).

DF. If f (x) is defined on the spectrum of the matrix A, then f (a) \u003d g (a), where G (a) is a polynomial that takes the same meanings on the spectrum as f (a),

DF. The value of the function from the matrix A Let's call the value of the polynomial from this matrix at.

Among the polynomials from [x] accepting the same values \u200b\u200bon the matrix spectrum, as, and F (x), the degree is not higher (M-1), receiving the same values \u200b\u200bon the spectrum, as well as f (x) is the balance of any division of any A polynomial G (x) having the same values \u200b\u200bon the matrix spectrum A, as well as f (x), to the minimum polynomial m (x) \u003d g (x) \u003d m (x) * g (x) + r (x).

This polynomial R (x) is called the interpolation polynomial of Lagrange Sylvester for the function f (x) on the spectrum of the matrix A.

Comment. If the minimum polynomial M (x) matrix is \u200b\u200bnot multiple roots, i.e. The value of the function on the spectrum.

Example:

Find R (x) for arbitrary f (x) if the matrix

. We construct f (H1 ). We find the minimum polynomial H1 Last invariant multiplier:

, D.n-1.\u003d X.2 ; D.n-1.=1;

m.x.\u003d F.n.(x) \u003d Dn.(x) / dn-1.(x) \u003d xn. 0 N.multiple root M (x), i.e. N-multiple Hi-values \u200b\u200bH1 .

, R (0) \u003d F (0), R(0) \u003d F(0), ..., R(N-1)(0) \u003d F(N-1)(0) .

1. Properties of functions from matrices.

Property number 1. If the matrix has eigenvalues \u200b\u200b(they can also include multiple), and, then the eigenvalues \u200b\u200bof the matrix F (a) are the eigenvalues \u200b\u200bof the polynomial F (X) :.

Evidence:

Let the characteristic polynomial of the matrix A takes:

Calculate. We turn from equality to the identifiers:

We will replace in equality:

Equality (*) is valid for any set F (x), so I will replace the polynomial F (X) on, we obtain:

On the left, we obtained the characteristic polynomial for the matrix F (a), the laid out to the linear factors, from where it follows that the eigenvalues \u200b\u200bof the matrix F (a).

Scold.

Property number 2. Let the matrix and eigenvalues \u200b\u200bof the matrix A, F (X) arbitrary function defined on the spectrum of the matrix A, then the eigenvalues \u200b\u200bof the matrix F (a) are equal.

Evidence:

Because The function f (x) is defined on the spectrum of the matrix A, then there is an interpolation polynomial of the matrix R (x) of such that, and then f (a) \u003d r (a), and the matrix R (a) will be their own values \u200b\u200bby property number 1 will be which are equal accordingly.

Scold.

Property number 3. If a and in such matrices, i.e. , and f (x) an arbitrary function defined on the spectrum of the matrix A, then

Evidence:

Because A and B are similar, their characteristic polynomials are the same and their own values \u200b\u200bare the same, so the value f (x) on the spectrum of the matrix A coincides with the value of the function f (x) on the spectrum of the matrix in, with the interpolation polynomial R (x), such that f (a) \u003d r (a) ,.

Scold.

Property number 4. If a block-diagonal matrix, then

Consequence: If, where F (x) the function defined on the spectrum of the matrix A.

1. Interpolation polynomial Lagrange Sylvester.

Case number 1.

Let it be given. Consider the first case: the characteristic polynomial has exactly n roots, among which there are no multiple, i.e. All eigenvalues \u200b\u200bof the matrix are different, i.e. , SP A simple. In this case, we construct the basic polynomials LK (X):

Let F (x) be a function defined on the matrix spectrum A and the values \u200b\u200bof this function on the spectrum will be. Need to build.

Build:

Pay attention that.

Example: Build an interpolation polynomial Lagrange Sylvester for the matrix.

Build basic polynomials:

Then for the function f (x), determined on the spectrum of the matrix A, we will get:

Take, then interpolation polynomial

Case number 2.

The characteristic polynomial matrix A has multiple roots, but the minimum polynomial of this matrix is \u200b\u200ba divider of the characteristic polynomial and has only simple roots, i.e. . In this case, the interpolation polynomial is built as in the previous case.

Case number 3.

Consider a general case. Let the minimum polynomial looks like:

where m1 + m2 + ... + ms \u003d m, deg r (x)

Let's make a fractional rational function:

and lay it on the simplest fraction.

Denote:. Multiply (*) on and get

where some function that does not appeal to infinity at.

If in (**) put, we get:

In order to find AK3 it is necessary (**) to properly delete twice, etc. Thus, the AKI coefficient is definitely determined.

After finding all the coefficients will return to (*), multiplying on m (x) and we obtain the interpolation polynomial R (x), i.e.

Example: Find F (A) ifwhere T. some parameter

Check whether the function is defined on the spectrum of the matrix A

Multiply (*) on (x-3)

at x \u003d 3

Multiply (*) on (x-5)

In this way, - Interpolation polynomial.

Example 2.

If athen prove that

We find the minimum polynomial matrix A:

- characteristic polynomial.

d.2 (x) \u003d 1, then the minimum polynomial

Consider f (x) \u003d sin x on the matrix spectrum:

The function is defined on the spectrum.

Multiply (*) on

.

Multiply (*) on:

Calculate, taking a derivative (**):

. Believed,

..

So,,

Example 3.

Let F (x) be defined on the matrix spectrum, the minimum polynomial of which is. Find the interpolation polynomial R (x) for the function f (x).

Solution: under the condition F (X) is defined on the spectrum of the matrix A F (1), F(1), f (2), f(2), f (2) Defined.

We use the method of indefinite coefficients:

If f (x) \u003d ln x

f (1) \u003d 0f.(1)=1

f (2) \u003d ln 2f.(2)=0.5 f.(2)=-0.25

4. Simple matrices.

Let the matrix, because with an algebraically closed field, then

The matrix analysis or the matrix method was widely distributed in comparative assessment of various business systems (enterprises, individual units of enterprises, etc.). The matrix method allows you to determine the integral assessment of each enterprise in several indicators. This assessment is called the rating of the enterprise. Consider the use of the matrix method of stages on a specific example.

1. Selection of estimated indicators and formation of the initial data matrix A IJ, that is, the tables where the rows are reflected in the system numbers (enterprises), and on the columns of the number numbers (i \u003d 1.2 ... .n) - systems; (J \u003d 1.2 ... ..n) - indicators. The selected indicators must have the same orientation (the more, the better).

2. Drawing up the matrix of standardized coefficients. In each column, the maximum element is determined, and then all elements of this column are divided into the maximum element. According to the calculation, the matrix of standardized coefficients is created.

We allocate the maximum element in each column.

Course of lectures on discipline

"Matrix Analysis"

for students II students

mathematical faculty specialty

"Economic cybernetics"

(lecturer Dmitruk Maria Aleksandrovna)

Chapter 3. Functions from matrices.

1. Definition of function.

DF. Let - The function of the scalar argument. It is required to determine what to understand under F (a), i.e. It is necessary to extend the function f (x) to the matrix value of the argument.

The solution of this problem is known when F (X) is a polynomial:, then.

Definition F (A) in the general case.

Let M (x) be the minimum polynomial A and it has such a canonical decomposition, - Own values \u200b\u200bA. Let the polynomials G (x) and h (x) take the same values.

Let G (a) \u003d h (a) (1) be (1), then the polynomial D (x) \u003d g (x) -h (x) is the annuling polynomial for A, since D (a) \u003d 0, therefore, D (x ) It is divided into a linear polynomial, i.e. d (x) \u003d m (x) * Q (x) (2).

Then, i.e. (3), , , .

We agree M numbers for f (x) such to call the values \u200b\u200bof the function f (x) on the spectrum of the matrix A, and the set of these values \u200b\u200bwill be denoted.

If the set F (SP A) is defined for f (x), the function is determined on the spectrum of the matrix A.

From (3) it follows that the polynomials h (x) and g (x) have the same values \u200b\u200bon the spectrum of the matrix A.

Our reasoning is reversible, i.e. From (3) þ (3) þ (1). Thus, if the matrix A is specified, then the value of the polynomial F (x) is determined by the values \u200b\u200bof this polynomial on the spectrum of the matrix A, i.e. All polynomials G i (x), receiving the same values \u200b\u200bon the matrix spectrum have the same matrix values \u200b\u200bG i (a). We will require that the definition of the value F (a) in the general case submitted to the same principle.

The values \u200b\u200bof the function f (x) on the spectrum of the matrix A should be fully determined f (a), i.e. The functions having the same values \u200b\u200bon the spectrum should have the same matrix value F (a). It is obvious that to determine F (a), in general, it is sufficient to find a polynomial G (x), which would take the same values \u200b\u200bon the spectrum A, as well as the function f (a) \u003d G (a).

DF. If f (x) is defined on the spectrum of the matrix A, then f (a) \u003d G (a), where G (a) is a polynomial that takes the same meanings on the spectrum as f (a),

DF. The value of the function from the matrix and we call the value of the polynomial from this matrix when .

Among the polynomials from [x], taking the same values \u200b\u200bon the matrix spectrum, as, and F (x), the degree is not higher (M-1), which takes the same values \u200b\u200bon the spectrum A, and F (X) is the balance of division Any polynomial G (x) having the same values \u200b\u200bon the matrix spectrum, as well as f (x), to the minimum polynomial M (x) \u003d G (x) \u003d m (x) * G (x) + R (x) .

This polynomial R (x) is called the interpolation polynomial of Lagrange Sylvester for the function f (x) on the spectrum of the matrix A.

Comment. If the minimum polynomial M (x) matrix is \u200b\u200bnot multiple roots, i.e. The value of the function on the spectrum.

Find R (x) for arbitrary f (x) if the matrix

. We construct F (H 1). We find the minimum polynomial H 1 - the last invariant multiplier:

, D n - 1 \u003d x 2; D n - 1 \u003d 1;

m x \u003d f n (x) \u003d d n (x) / d n-1 (x) \u003d x n þ 0 - N -Chent root M (x), i.e. N-multiple eigenvalues \u200b\u200bH 1.

R (0) \u003d f (0), R '(0) \u003d F' (0), ..., R (n - 1) (0) \u003d f (n - 1) (0) þ.

Troika is a game decision<=>When a game is solved, where a is any real number, K\u003e 0 Chapter 2. Games with a zero amount in net strategies 2.1 Calculation of optimal strategies on the example of solving problems using the minimax theorem, it can be argued that each antagonistic game has optimal strategies. Theorem: Let A - a matrix game and strings of this ...

The picture that does not correspond to it are candidates for an exception from the scope of the corporation. 5. Development of a corporate strategy Preceding analysis has prepared the soil to develop strategic steps to improve the activities of a diversified company. The main conclusion about what to do depends on the conclusions relating to the whole set of activities in the economic activity ...