Course of lectures on the discipline
"Matrix Analysis"
for 2nd year students
Faculty of Mathematics specialty
"Economic cybernetics"
(lecturer Dmitruk Maria Alexandrovna)
Chapter 3. Functions of matrices.
- Definition of a function.
Df. Let the function be a scalar argument. It is necessary to determine what is meant by f(A), i.e. you need to extend the function f(x) to the matrix value of the argument.
The solution to this problem is known when f(x) is a polynomial: , then.
Definition of f(A) in the general case.
Let m(x) be a minimal polynomial of A and it has such a canonical expansion, eigenvalues of A. Let the polynomials g(x) and h(x) take the same values.
Let g(A)=h(A) (1), then the polynomial d(x)=g(x)-h(x) is a annihilating polynomial for A, since d(A)=0, therefore d(x) is divided by a linear polynomial, i.e. d(x)=m(x)*q(x) (2).
Then, i.e. (3), .
Let us agree to call m numbers for f(x) the values of the function f(x) on the spectrum of matrix A, and we will denote the set of these values.
If the set f(Sp A) is defined for f(x), then the function is defined on the spectrum of the matrix A.
From (3) it follows that the polynomials h(x) and g(x) have the same values on the spectrum of matrix A.
Our reasoning is reversible, i.e. from (3) (3) (1). Thus, if matrix A is given, then the value of the polynomial f(x) is completely determined by the values of this polynomial on the spectrum of matrix A, i.e. all polynomials gi(x) taking the same values on the spectrum of the matrix have the same matrix values gi(A). We require that the determination of the value of f(A) in the general case obey the same principle.
The values of the function f(x) on the spectrum of matrix A must fully determine f(A), i.e. functions that have the same values on the spectrum must have the same matrix value f(A). Obviously, to determine f(A) in the general case, it is enough to find a polynomial g(x) that would take the same values on the spectrum A as the function f(A)=g(A).
Df. If f(x) is defined on the spectrum of matrix A, then f(A)=g(A), where g(A) is a polynomial that takes the same values on the spectrum as f(A),
Df. The value of the function from matrix A let's call the value of the polynomial from this matrix at.
Among the polynomials from C[x], taking the same values on the spectrum of the matrix A, as f(x), the degree is not higher than (m-1), taking the same values on the spectrum A, as f(x), this is the remainder of the division of any polynomial g(x), which has the same values on the spectrum of matrix A as f(x), to the minimal polynomial m(x)=g(x)=m(x)*g(x)+r(x).
This polynomial r(x) is called the Lagrange-Sylvester interpolation polynomial for the function f(x) on the spectrum of the matrix A.
Comment. If the minimal polynomial m(x) of matrix A does not have multiple roots, i.e. , then the value of the function on the spectrum.
Example:
Find r(x) for an arbitrary f(x), if the matrix
. Let us construct f(H1 ). Let's find the minimal polynomial H1 last invariant factor:
,dn-1=x2 ; dn-1=1;
mx=fn(x)=dn(x)/dn-1(x)=xn 0 nmultiple root of m(x), i.e. n-fold eigenvalues H1 .
, r(0)=f(0), r(0)=f(0),…,r(n-1)(0)=f(n-1)(0) .
- Properties of functions from matrices.
Property No. 1. If the matrix has eigenvalues (among them there may be multiples), a, then the eigenvalues of the matrix f(A) are the eigenvalues of the polynomial f(x): .
Proof:
Let the characteristic polynomial of matrix A have the form:
Let's do the math. Let's move from equality to determinants:
Let's make a replacement in equality:
Equality (*) is true for any set f(x), so we replace the polynomial f(x) with, we get:
On the left we have obtained the characteristic polynomial for the matrix f(A), decomposed on the right into linear factors, which implies that the eigenvalues of the matrix f(A).
CTD.
Property No. 2. Let the matrix and the eigenvalues of the matrix A, f(x) be an arbitrary function defined on the spectrum of the matrix A, then the eigenvalues of the matrix f(A) are equal.
Proof:
Because function f(x) is defined on the spectrum of the matrix A, then there is an interpolation polynomial of the matrix r(x) such that, and then f(A)=r(A), and the eigenvalues of the matrix r(A) by property No. 1 will be which are respectively equal.
CTD.
Property No. 3. If A and B are similar matrices, i.e. , and f(x) is an arbitrary function defined on the spectrum of matrix A, then
Proof:
Because A and B are similar, then their characteristic polynomials are the same and their eigenvalues are the same, therefore the value of f(x) on the spectrum of matrix A coincides with the value of the function f(x) on the spectrum of matrix B, and there is an interpolation polynomial r(x) such that that f(A)=r(A), .
CTD.
Property No. 4. If A is a block diagonal matrix, then
Consequence: If, then where f(x) is a function defined on the spectrum of matrix A.
- Lagrange-Sylvester interpolation polynomial.
Case No. 1.
Let it be given. Let's consider the first case: the characteristic polynomial has exactly n roots, among which there are no multiples, i.e. all eigenvalues of matrix A are different, i.e. , Sp A simple. In this case, we construct the basis polynomials lk(x):
Let f(x) be a function defined on the spectrum of matrix A and let the values of this function on the spectrum be. We need to build it.
Let's build:
Let us note that.
Example: Construct a Lagrange-Sylvester interpolation polynomial for a matrix.
Let's construct basic polynomials:
Then for the function f(x), defined on the spectrum of matrix A, we obtain:
Let's take, then the interpolation polynomial
Case No. 2.
The characteristic polynomial of matrix A has multiple roots, but the minimal polynomial of this matrix is a divisor of the characteristic polynomial and has only simple roots, i.e. . In this case, the interpolation polynomial is constructed in the same way as in the previous case.
Case No. 3.
Let's consider the general case. Let the minimal polynomial have the form:
where m1+m2+…+ms=m, deg r(x) Let's create a fractional rational function: and break it down into simple fractions. Let's denote: . Multiply (*) by and get where is some function that does not go to infinity at. If we put it in (**), we get: In order to find ak3 you need to (**) differentiate twice, etc. Thus, the coefficient aki is determined uniquely. After finding all the coefficients, we return to (*), multiply by m(x) and obtain the interpolation polynomial r(x), i.e. Example: Find f(A) if, where tsome parameter
Let's check whether the function is defined on the spectrum of matrix A Multiply (*) by (x-3) at x=3 Multiply (*) by (x-5) Thus,- interpolation polynomial. Example 2. If, then prove that Let's find the minimum polynomial of matrix A: - characteristic polynomial. d2
(x)=1, then the minimum polynomial Consider f(x)=sin x on the spectrum of the matrix: the function is defined on the spectrum. Multiply (*) by
.
Multiply (*) by:
Let's calculate by taking the derivative (**): . Believing,
, i.e..
So,,
Example 3. Let f(x) be defined on the spectrum of a matrix whose minimal polynomial has the form. Find the interpolation polynomial r(x) for the function f(x). Solution: By condition, f(x) is defined on the spectrum of matrix A f(1), f(1), f(2), f(2), f(2) defined. We use the method of undetermined coefficients: If f(x)=ln x f(1)=0f(1)=1
f(2)=ln 2f(2)=0.5
f(2)=-0.25
4. Simple matrices.
Let be a matrix, since C is an algebraically closed field, then Matrix analysis or the matrix method has found widespread use in the comparative assessment of various economic systems (enterprises, individual divisions of enterprises, etc.). The matrix method allows you to determine the integral assessment of each enterprise based on several indicators. This assessment is called the enterprise rating. Let us consider the application of the matrix method step by step using a specific example. 1. Selection of evaluation indicators and formation of the initial data matrix a ij, that is, tables where the rows reflect the numbers of systems (enterprises), and the columns reflect the numbers of indicators (i=1,2....n) - the systems; (j=1,2…..n) - indicators. The selected indicators must have the same focus (the more, the better). 2. Drawing up a matrix of standardized coefficients. In each column, the maximum element is determined, and then all elements in that column are divided by the maximum element. Based on the calculation results, a matrix of standardized coefficients is created. Select the maximum element in each column. Historically, the first model of corporate strategic planning is considered to be the so-called “growth-share” model, which is better known as the Boston Consulting Group (BCG) model. This model is a kind of display of the positions of a particular type of business in a strategic space defined by two axes (x, y), one of which is used to measure the growth rate of the market for the corresponding product, and the other to measure the relative share of the organization’s products in the market for the product in question. The emergence of the BCG model was the logical conclusion of a research work conducted at one time by a specialist from the consulting company Boston Consulting Group. In the process of studying various organizations producing 24 main types of products in 7 industries (electric power, plastics production, non-ferrous metals industry, electrical equipment production, gasoline production, etc.), empirical facts were established that when production volumes double, variable production costs units of production are reduced by 10-30%. It has also been found that this trend occurs in almost every market sector. These facts became the basis for the conclusion that variable production costs are one of the main, if not the main, factor of business success and determines the competitive advantages of one organization over another. Using statistical methods, empirical dependencies were derived that describe the relationship between production costs, units of production and production volume. And one of the main factors of competitive advantage was placed in unambiguous correspondence with the volume of production, and, consequently, with the share of the market for the corresponding products that this volume occupies. The main focus of the BCG model is on the cash flow of an enterprise, which is directed either to conduct operations in a particular business area, or arises as a result of such operations. It is believed that the level of income or cash expenditure is very strongly functionally dependent on the growth rate of the market and the relative share of the organization in this market. The growth rate of an organization's business determines the rate at which the organization will use cash. It is generally accepted that during the maturity stage and the final stage of the life cycle of any business, a successful business generates cash, while during the development and growth stage of a business, cash is consumed. Conclusion: To maintain the continuity of a successful business, the money supply resulting from the implementation of a “mature” business must be partially invested in new areas of business that promise to become generators of income for the organization in the future. In the BCG model, the main commercial goals of the organization are assumed to be growth in mass and profit margins. At the same time, the set of acceptable strategic decisions regarding how these goals can be achieved is limited to 4 options: The decisions that the BCG model suggests depend on the position of the organization’s specific type of business, the strategic space formed by two coordinate axes. The use of this parameter in the BCG model is possible for 3 reasons: a growing market, as a rule, promises a return on investment in this type of business in the near future. increased market growth rates affect the amount of cash with a “-” sign, even in the case of a fairly high rate of profit, since it requires increased investment in business development. There are two BCG models: classic and adapted. Consider the Classic model: Structure of the Classic Model: The x-axis shows a measurement of some of the organization's competitive positions in a given business in the form of the ratio of the organization's sales volumes in a given business to the sales volume of the largest competitor in a given business area. In the original BCG version, the abscissa scale is logarithmic. Thus, the BCG model is a 2 * 2 matrix on which business areas are displayed by circles with centers at the intersection of coordinates formed by the corresponding market growth rates and the relative share of the organization in the corresponding market. Each drawn circle characterizes only 1 business area characteristic of a given organization. The size of the circle is proportional to the total size of the entire market. Most often, this size is determined by simply adding the organization's business and the corresponding business of its competitors. Sometimes a segment is identified on each circle that characterizes the relative share of the organization’s business area in a given market, although this is not necessary to obtain strategic conclusions in this model. The division of the axes into 2 parts was not done by chance. At the top of the matrix are business areas with above-average growth rates. At the bottom, correspondingly lower. The original BCG model assumed that the boundary between high and low growth rates was a 10% increase in sales per year. Each of these squares is given figurative names (for example: the BCG matrix is called the “Zoo”). “Stars”: these are new business areas that occupy a relatively large share of a rapidly developing market in which they generate high profits. These business areas can be called leaders in their industries, since they bring very high income to the organization. However, the main challenge lies in determining the right balance between income and investment in this area in order to ensure the return of the latter in the future. Cash Cows: These are business areas that have gained a relatively large market share in the past, but over time the growth of the respective industry has slowed down noticeably, the cash flow in this position is well balanced since the bare minimum is required to invest in such a business area. Such a business area can bring good income to the organization (These are former “Stars”). Problem Children: These business areas compete in growing industries but have a relatively small market share. This combination of circumstances leads to the need to increase investment in order to protect its market share. High growth rates require significant cash flow to keep up with that growth. "Dogs": These are business areas with a relatively small market share in slow-growing industries. Cash flow is negligible, sometimes even negative. But not many people use the Classic model, since it is impractical due to the need to obtain up-to-date data on the state of the market and the share occupied by the company and its competitor. Therefore, for calculations we use Adapted model: The adapted BCG matrix is built on the basis of internal company information. Required data - product sales volumes for a certain period, which cannot be less than 12 months; in the future, to track dynamics, it is necessary to add data for the next 3 months (i.e. data for 12, 15, 18, 21, 24 months) . The data does not have to start with the month of January, but should be by month. It is also important to consider the seasonality of sales of goods or services for your company's products. In the company under consideration, the product portfolio consists of 5 groups of goods, and there is also data on their sales for the period January - December 2013. Table 5. Sales data for NordWest LLC Ceramic tile Cellular concrete Large format brick – by multiplying the weight by the assessment and summing up the obtained values for all factors, we obtain a weighted assessment / rating of market attractiveness Table 7. Industry attractiveness assessment Table 8. Assessment of competitive position in the industry 2
.We are building the McKinsey Matrix for Nord-West LLC On the x-axis we plot 3.6 points, on the y-axis we plot 2.9 points. At the intersection of these points we find ourselves in the “Success 3” square. Which is inherent in organizations whose market attractiveness is at an average level, but at the same time their advantages in this market are obvious and strong. The strategic conclusions from the analysis based on the McKinsey matrix are obvious: the company Nord-West LLC “falls into the “Success 3” square Rice. 4. McKinsey Matrix
The “success 3” position is characterized by the highest degree of market attractiveness and relatively strong advantages in it. The company will be the undisputed leader or one of the leaders in the construction market, and the threat to it can only be the strengthening of some positions of individual competitors. Therefore, the strategy of an enterprise that is in such a position should be aimed at protecting its wealth, mostly through additional investments. An organization needs, first of all, to identify the most attractive market segments and invest in them, develop its advantages and resist the influence of competitors. Course of lectures on the discipline "Matrix Analysis"
for 2nd year students
Faculty of Mathematics specialty
"Economic cybernetics"
(lecturer Dmitruk Maria Alexandrovna)
1.
Definition of a function.
Df. Let The solution to this problem is known when f(x) is a polynomial: Definition of f(A) in the general case.
Let m(x) be the minimal polynomial A and it has the following canonical expansion Let g(A)=h(A) (1), then the polynomial d(x)=g(x)-h(x) is a canceling polynomial for A, since d(A)=0, therefore d(x ) is divided by a linear polynomial, i.e. d(x)=m(x)*q(x) (2). Let us agree on m numbers for f(x) such If the set f(Sp A) is defined for f(x), then the function is defined on the spectrum of the matrix A. From (3) it follows that the polynomials h(x) and g(x) have the same values on the spectrum of matrix A. Our reasoning is reversible, i.e. from (3) Þ (3) Þ (1). Thus, if matrix A is given, then the value of the polynomial f(x) is completely determined by the values of this polynomial on the spectrum of matrix A, i.e. all polynomials g i (x) taking the same values on the spectrum of the matrix have the same matrix values g i (A). We require that the determination of the value of f(A) in the general case obey the same principle. The values of the function f(x) on the spectrum of matrix A must fully determine f(A), i.e. functions that have the same values on the spectrum must have the same matrix value f(A). Obviously, to determine f(A) in the general case, it is enough to find a polynomial g(x) that would take the same values on the spectrum A as the function f(A)=g(A). Df. If f(x) is defined on the spectrum of matrix A, then f(A)=g(A), where g(A) is a polynomial that takes the same values on the spectrum as f(A), Df.The value of the function from matrix A
let's call the value of the polynomial from this matrix at Among the polynomials from C[x], taking the same values on the spectrum of the matrix A, as f(x), the degree is not higher than (m-1), taking the same values on the spectrum A, as f(x) - this is the remainder of the division any polynomial g(x) having the same values on the spectrum of matrix A as f(x), to the minimal polynomial m(x)=g(x)=m(x)*g(x)+r(x) . This polynomial r(x) is called the Lagrange-Sylvester interpolation polynomial for the function f(x) on the spectrum of the matrix A. Comment.
If the minimal polynomial m(x) of matrix A does not have multiple roots, i.e. Example: Find r(x) for an arbitrary f(x), if the matrix . Let us construct f(H 1). Let's find the minimal polynomial H 1 - the last invariant factor: , d n-1 =x 2 ; d n-1 =1; m x =f n (x)=d n (x)/d n-1 (x)=x nÞ
0 – n-fold root m(x), i.e. n-fold eigenvalues of H 1 . , r(0)=f(0), r’(0)=f’(0),…,r (n-1) (0)=f (n-1) (0)Þ
.
2.
Properties of functions from matrices.
Property No. 1.
If matrix Proof:
Let the characteristic polynomial of matrix A have the form: Let's make a replacement in equality: Equality (*) is true for any set f(x), so we replace the polynomial f(x) with On the left we have obtained the characteristic polynomial for the matrix f(A), decomposed on the right into linear factors, which implies that CTD.
Property No. 2.
Let the matrix Proof:
Because function f(x) is defined on the spectrum of the matrix A, then there is an interpolation polynomial of the matrix r(x) such that In strategic planning and marketing, quite a lot of matrices of one direction or another are used. There is a need to systematize these matrices, as well as the gradual implementation of the matrix approach at all stages of strategic analysis and planning. Levels of strategic planning in matrix measurement. In strategic planning, one can distinguish the corporate level, the business level, and the functional level. Strategic planning matrices at the corporate level analyze the businesses included in the corporation, i.e. help to carry out portfolio analysis, as well as analysis of the situation in the corporation as a whole. The business level includes matrices that are relevant to a given business unit. Matrices most often refer to one product, analyze the properties of this product, the situation on the market for this product, etc. Functional level matrices examine factors influencing the functional areas of the enterprise, of which the most important are marketing and personnel. Existing strategic analysis and planning matrices examine various aspects of this process. Classification of matrices is necessary to identify patterns and features of the use of the matrix method in strategic analysis and planning. Matrices can be classified according to existing characteristics as follows: The more cells a matrix contains, the more complex and informative it is. In this case, it is possible to divide the matrices into four groups. The first group includes matrices consisting of four cells. The second group contains matrices consisting of nine cells, the third - sixteen, and the fourth - more than sixteen cells. Classification by object of study divides matrices into groups depending on the object being studied. In the “Awareness - Attitude” matrix, the object of study is the personnel, as well as in the “Impact of payment on group relationships” matrix. Another object of study is the company's portfolio. Examples in this group include Shell/DPM and BCG matrices. This classification divides matrices into two groups according to the information received: either quantitative or semantic. In this group, an example of a matrix formed by information in the form of a number is the matrix of the vector of the economic state of the organization, and formed by logical information is the matrix of the main forms of associations. At the first stage, it is proposed to carry out a primary analysis of the enterprise’s activities. Three matrices have been selected for this purpose. The SWOT matrix is widely described in the literature. The MCC matrix involves analyzing the alignment of the enterprise's mission and its core capabilities. The matrix of the vector of economic development of an enterprise is a table that presents numerical data of the main indicators of the enterprise. From this matrix you can glean information for other matrices, and based on this data, you can draw various conclusions already at this stage. The second stage of applying matrix methods is market and industry analysis. Here the markets in which the enterprise operates, as well as the industry as a whole, are analyzed. The main ones in the “Market” subgroup are the BCG matrix, which examines the relationship between growth rates and market share, and the GE matrix, which analyzes the comparative attractiveness of the market and competitiveness in the industry and has two varieties: the Day version and the Monienson version. The “Industry” subgroup contains matrices that study the industry environment and patterns of industry development. The main one in this subgroup is the Shell/DPM matrix, which studies the relationship between industry attractiveness and competitiveness. The next stages of strategic planning are differentiation analysis and quality analysis. Differentiation and quality act in this case as components with the help of which it is possible to obtain the required result. There are three matrices in the “Differentiation” group. The “Improving Competitive Position” matrix allows you to clearly identify patterns and dependencies of differentiation on market coverage. The matrix “Differentiation - relative cost efficiency” reveals the dependence of the relative cost efficiency in a given market on differentiation. The Performance-Innovation/Differentiation matrix shows the relationship between the performance of a given business unit and the implementation of innovation. The object of research of the “Quality Analysis” group is the identification of factors and patterns that influence such an aspect as the quality of manufactured products. A group can include two matrices. The Pricing Strategies matrix positions products based on quality and price. The “Quality – Resource Intensity” matrix determines the ratio of the quality of the product produced and the resources spent on it. The “Management Analysis” and “Marketing Strategy Analysis” groups are not included in the process of step-by-step implementation of the matrix method in strategic planning. These groups are separate. The matrices that make up these groups can be used at all stages of strategic planning and address issues of functional planning. The Management Analysis group consists of two subgroups. The first subgroup - “Management” - examines the management of the company as a whole, the processes affecting the management, and management of the company. The “Personnel” subgroup examines the processes occurring between colleagues and the influence of various factors on the performance of personnel. In the proposed scheme of strategic analysis and planning in each group, the matrices interact with each other, but you cannot rely on the result or conclusion of only one matrix - it is necessary to take into account the conclusions obtained from each matrix in the group. After the analysis is carried out in the first group, the analysis is carried out in the next one. Analysis in the “Management” and “Marketing Strategy” groups is carried out at all stages of analysis in strategic planning. SWOT analysis is one of the most common types of analysis in strategic management today. SWOT: Strengths; Weaknesses; Opportunities; Threats. SWOT analysis allows you to identify and structure the strengths and weaknesses of a company, as well as potential opportunities and threats. This is achieved by comparing the internal strengths and weaknesses of their company with the opportunities that the market gives them. Based on the quality of compliance, a conclusion is made about the direction in which the business should develop, and ultimately the distribution of resources among segments is determined. The purpose of SWOT analysis is to formulate the main directions of development of an enterprise through systematization of available information about the strengths and weaknesses of the company, as well as potential opportunities and threats. The most attractive thing about this method is that the information field is formed directly by the managers themselves, as well as by the most competent employees of the company, based on the generalization and coordination of their own experience and vision of the situation. A general view of the primary SWOT analysis matrix is shown in Fig. 1. Fig.1. Matrix of primary strategic SWOT analysis. Based on a consistent consideration of factors, decisions are made to adjust the goals and strategies of the enterprise (corporate, product, resource, functional, managerial), which, in turn, determine the key points of organizing activities. Analysis of a company's business portfolio should help managers assess the company's field of activity. The company should strive to invest in more profitable areas of its activities and reduce unprofitable ones. The first step for management when analyzing a business portfolio is to identify the key areas of activity that define the company's mission. They can be called strategic business elements - SEB. In the next stage of business portfolio analysis, management must assess the attractiveness of the various SSEs and decide how much support each deserves. In some companies this happens informally during the work process. Management examines the company's portfolio of activities and products and, using common sense, decides how much each SEB should bring in and receive. Other companies use formal methods for portfolio planning. Formal methods can be called more accurate and thorough. Among the most well-known and successful methods of analyzing a business portfolio using formal methods are the following: The BCG method is based on the principle of analyzing the growth/market share matrix. This is a portfolio planning method that evaluates a company's SEB in terms of their market growth rate and the relative share of those elements in the market. SEBs are divided into “stars”, “cash cows”, “dark horses” and “dogs” (see Fig. 2). R R Fig.2. BCG Matrix. The vertical axis in Fig. 2, market growth rate, determines the measure of market attractiveness. The horizontal axis, relative market share, determines the strength of a company's position in the market. When dividing the growth/market share matrix into sectors, four types of EBS can be distinguished. "Stars". Rapidly developing areas of activity, products with a large market share. They usually require heavy investment to maintain their growth. Over time, their growth slows down and they turn into “cash cows”. "Cash cows" Lines of business or products with low growth rates and large market shares. These sustainable, successful SEBs require less investment to maintain their market share. At the same time, they generate high income, which the company uses to pay its bills and to maintain other self-assessment systems that require investment. "Dark horses". Elements of a business that have a small share of high-growth markets. They require a lot of capital to even maintain their market share, let alone increase it. Management should carefully consider which dark horses should be turned into stars and which should be phased out. "Dogs". Business lines and products with low growth rates and small market shares. They may generate enough income to support themselves, but do not promise to become more serious sources of income. Each SEB is placed on this matrix in proportion to its share in the company’s gross income. After classifying the EBS, the company must determine the role of each element in the future. For each SEB, one of four strategies can be applied. A company may increase investment in an element of its business to gain market share for it. Or it can invest exactly as much as is necessary to maintain the SEB share at the current level. It can pump resources out of the SEB, withdrawing its short-term monetary resources for a certain period of time, regardless of long-term consequences. Finally, it can divest from the SEB by selling it or phasing it out and use the resources elsewhere. Over time, SEB changes its position in the growth/market share matrix. Each SEB has its own life cycle. Many SEBs start out as “dark horses” and, under favorable circumstances, move into the category of “stars”. Later, as market growth slows, they become “cash cows” and, finally, at the end of their life cycle, they fade away or turn into “dogs.” The company needs to continuously introduce new products and activities so that some of them become “stars” and then “cash cows” that help finance other SEBs. Matrix methods play a very important role in strategic analysis, planning and marketing. The matrix method is very convenient - this is what explains its prevalence. However, using only matrix methods is not sufficient, since matrices allow you to study strategic planning and marketing from individual aspects, and do not show the full picture, but in combination with other methods, the matrix approach makes it possible to clearly see patterns in the processes occurring in the enterprise and make correct conclusions. Table 1. Matrix tools in the analysis and planning of organizational activities
If you notice an error in the text, highlight the word and press Shift + Enter
Classification of strategic analysis and planning matrices.
Introduction of matrix tools into the analysis and planning of enterprise activities.
Characteristics of individual matrices
T
e
m
P
O
With
T
A
s
n
To
AV
s
With
O
To
And
th"Star"
"Cash cows"
n
And
h
To
And
th"Milch cow"
"Dog"
high
low
Relative market share
№
Levels of problem solving
Matrix
Main characteristics
1
Primary analysis
SWOT Matrix
Analysis of the strengths and weaknesses of the enterprise, opportunities and threats
2
MCC Matrix
Analysis of compliance with the mission of the enterprise and its main capabilities
3
Matrix of the vector of economic development of an enterprise
Analysis of statistical data
4
Market/Industry Analysis
BCG Matrix
Growth rate and market share analysis
5
GE Matrix
Analysis of comparative market attractiveness and competitiveness
6
ADL Matrix
Analysis of industry life cycle and relative market position
7
HoferSchendel Matrix
Analysis of the position among competitors in the industry and the stage of market development
8
Ansoff matrix
(“market-product”)Analysis of strategy in relation to markets and products
9
Porter Matrix
(five competitive forces)Analysis of strategic business development prospects
10
Elasticity matrix of competitive response in the market
Analysis of the company's actions on the factors of competitiveness of the product depending on the elasticity of the reaction of the priority competitor for the product
11
Product grouping matrix
Product grouping analysis
12
Matrix “Impact Uncertainty”
Analysis of the level of impact and degree of uncertainty when entering a new market
13
Industry
Cooper Matrix
Analysis of industry attractiveness and business strength
14
ShellDPM Matrix
Analysis of the attractiveness of a resource-intensive industry depending on competitiveness
15
Matrix of strategies for a declining business
Analysis of competitive advantages in the industry environment
16
Matrix of basic forms of associations
Analysis of mergers in an industry environment
17
Differentiation Analysis
Competitive Position Improvement Matrix
Market differentiation and coverage analysis
18
Matrix “Differentiation relative cost effectiveness”
Analysis of differentiation and relative cost-effectiveness
19
Matrix “Performance - Innovation/Differentiation”
Innovation/Differentiation and Productivity Analysis
20
Quality Analysis
Matrix “Price-quality”
Product positioning based on quality and price
21
Matrix
“Quality - resource intensity”Analysis of the dependence of quality on resource intensity
22
Marketing strategy analysis
Brand family expansion strategy matrix
Analysis of the relationship between distinctive advantages and target market segmentation
23
Matrix “Awareness-attitude towards a product brand”
Analysis of the relationship between gross profit margin and sales response
24
Marketing Channel Matrix
Analysis of the relationship between the pace of market development and the value added by the channel
25
Matrix “Contact - level of service adaptation”
Analysis of the dependence of the level of adaptation of services to customer requirements on the degree of contact with the client
26
Matrix
“Marketing Diagnostics”Analysis of the dependence of strategy on strategy implementation
27
Management Analysis
ManagementMatrix of strategic management methods
Analysis of the relationship between strategy and the impact of planning
28
Strategic management model matrix
Analysis of the dependence of the management model on the type of change
29
Hersey-Blanchard matrix
Analysis of situational leadership model
30
Matrix “Dimensional Combinations of Ohio University Leadership Styles”
Analysis of combinations of leadership styles dimensions
31
Matrix “Management grid”
Analysis of leadership types
32
Staff
Matrix “Change - in the organization”
Analysis of the dependence of changes occurring in the organization and resistance to these changes
33
Matrix of the influence of payment on relationships in the group
Analysis of the dependence of relationships in the group on payment differentiation
34
Matrix of types of inclusion of a person in a group
Analysis of the relationship between attitude towards the values of the organization and attitude towards the norms of behavior in the organization
35
Matrix “Core Business Capabilities”
Analysis of the market and key business capabilities
36
Matrix “Importance of work”
Analysis of the dependence of work performance on importance
37
Matrix of existing formal systems of work quality criteria
Analysis of existing formal systems of work quality criteria
38
Performance Quality Criteria Management Results Matrix
Analysis of the results of managing work quality criteria
39
Blake-Mouton Matrix
Analysis of the dependence of work performance on the number of people and on the number of tasks
40
MacDonald Matrix
Performance Analysis
