6 semester


Name of the discipline



Course of Study

3, mathematics (economic activity)


Semester of training



Amount of credits



Full name of lecturer

Zabreiko Petr Petrovich


Objectives of studying the discipline

the formation of the fundamentals of mathematical thinking among students, mastering the methods of mathematical modeling of economic phenomena and processes, as well as the ability to interpret the results of research of mathematical models from the economic point of view.

As a result of studying the discipline, the trainee must

be able to:

– to compile mathematical models on economic data technological matrices;

– to check the productivity of technological matrices, to evaluate the effectiveness of the relevant models; determine positive and non-negative eigenvectors of technological matrices;

– calculate or evaluate the main characteristics of technological matrices;



Mathematical analysis

Probability theory

Functional Analysis


Contents of the discipline

Theme 1. Primitive and imprimitive matrices

Topic 2. Spectral radius estimates and their economic applications. Non-negative eigenvectors of indecomposable and decomposable matrices; main ones own. Calculation of the principal eigenvectors. Boolean matrices and their applications to the theory of nonnegative matrices. Theme 3. Theorems on perturbations of equilibrium. Hicks phenomenon in the open model of Leontief.

Theme 4. Iterative methods for studying the Leontief model. Birkhoff-Hilbert space. A theorem on the completeness of the Birkhoff-Hilbert space and the Birkhoff theorem. Computing a positive eigenvector. Spectral gap of positive matrices. Oscillation and Ostrovsky’s theorem. Bauer’s theorem. The geometric meaning of the theorems of Birkhoff, Ostrovsky, and Bauer. Estimates of the spectral gap. The Krasnoselsky-Sobolev theorem. Sufficient signs of primitivity of matrices. Sharp Matrices


Recommended literature

1. Allen R. Mathematical economics. – Moscow: Foreign Literature Publishing House, 1963. – 600 p.

2. Ashmanov S.A. Mathematical models and methods in economics. – Moscow: Publishing house of Moscow University, 1980. – 200 p.

3. Ashmanov S.A. Introduction to mathematical economics. – Moscow: Science, the main edition of physics and mathematics, 1984. – 294 p.

4. Voevodin VV, Kuznetsov Yu.A. Matrices and calculations

(Reference and Mathematical Library).} – M .: Nauka, The Main Edition of Physical and Mathematical Literature, 1984.- 320 p.

5. Gantmakher FR Theory of matrices. – Moscow: Science, Fizmatgiz, 1966. – 548 p.

6. Dorfman R., Samuelson P.A., Solow R.M. Linear Programming and Economic Analysis. – New York Toronto London: McGraw-Hill Book Company, Inc., 1958. – 525 p.

7. PP Zabreiko Mathematical Basics of Economics. – Lectures, 2002, 1-54.

8. Zabreiko PP, Shevelevich K.V. Theorems of Hicks and Le Chatelier-Samuelson for decomposable nonnegative matrices. – Reports of the National Academy of Sciences of Belarus, (2002), No 3, p. 30-34.

9. Intriligator M. Mathematical methods of optimization and economic theory. – Moscow: Progress, 1975. – 608 p.

10. Karlin S. Mathematical methods in the theory of games, programming and economics. – The World, 1964. – 839 p.

11. Kuhn HU, Tucker A.U. Linear inequalities and related questions. – M .: Publishing house of foreign literature, 1959. – 470 p.

12. Lancaster, K. Mathematical Economics. – Moscow: Soviet Radio, 1972. – 464 p.

13. Sydsetter K., A., Burke P. Handbook on

mathematics for economists. St. Petersburg: Economic School. St. Petersburg State University of Economics and Finance. Higher School of Economics, 2000. 1-230

14. Horn R., Johnson C. Matrix analysis. – Moscow: Mir,

1989, – 656 p. 15. Arrow K.D., Hurwitz L., Udzava H. Studies on linearity and nonlinear programming. – Moscow: The Publishing House of Foreign Literature, 1962. – 333 p.


Teaching Methods

interactive methods of teaching (working in small groups (team), problem training) are organized taking into account the inclusion in the learning process of all students of the group. Joint activity means that each student makes his own individual contribution, in the course of the work there is an exchange of knowledge, ideas, methods of activity. Interactive methods are based on the principles of interaction, activity of trainees, reliance on group experience, mandatory feedback


Language of education



Conditions (requirements), current control

– test;

– colloquium

The assessment on the exam is set taking into account: 30% – work in laboratory and practical classes, 70% – oral examination.


Form of certification