ON POSITIVE SOLUTIONS OF A NONHOMOGENEOUS SYSTEM OF LINEAR ALGEBRAIC EQUATIONS WITH A POSITIVE RIGHT-HAND SIDE

Авторы: 
Дата поступления: 
10.08.2017
Рубрика: 
Год: 
2017
Номер журнала (Том): 
УДК: 
519.61
DOI: 

10.26731/1813-9108.2017.3(55).22-30

Файл статьи: 
Страницы: 
22
30
Аннотация: 

The paper discusses the possibility of obtaining solutions with positive elements for a nonhomogeneous system of linear algebraic equations with a given quadratic matrix A and with an indefinite right-hand side under the assumption of positivity of their values. The investigation has been conducted on second and third order matrices. The two techniques of obtaining solutions have been proposed: the analytical technique and matrix technique.

The first technique has been developed for obtaining solutions of a system of inequalities. Its basis is comprised of elementary transformations, as a result of which some part of variables is removed from the system of inequalities, while simplifying analysis bound up with resolving the issue of existence of solutions for the system of inequalities. An algebraic criterion, which expresses necessary conditions of existence of solutions for the system of inequalities, has been proposed for this technique.

The second technique is based on principal properties of matrices: eigenvalues of matrices and eigenvectors. Sufficient conditions of existence of real solutions for the initial system of equations and sufficient conditions of impossibility of existence of such solutions have been obtained for this technique.

Список цитируемой литературы: 

1. Lyapunov A.M. Obshchaya zadacha ob ustoichivosti dvizheniya [The general problem of the stability of motion]. Vol. 2. Moscow-Leningrad: AS USSR Publ., 1956, pp. 7–263.

2. Chetaev N.G. Ustoichivost' dvizheniya; raboty po analiticheskoi mekhanike [Stability of motion; works on analytical mechanics]. Moscow: AS USSR Publ., 1962, 535 p.

3. Letov A.M. Ustoichivost' reguliruemykh system [Stability of regulated systems]. Moscow: Fizmatgiz Publ., 1962, 483 p.

4. Letov A.M. Matematicheskaya teoriya protsessov upravleniya [Mathematical theory of control processes]. Moscow: Nauka Publ., 1981, 255 p.

5. Zubov V.I. Ustoichivost' dvizheniya [Stability of motion]. Moscow: Vysshaya shkola Publ., 1963, 270 p.

6. Kamenkov V.G. Ustoichivost' dvizheniya; kolebaniya, aerodinamika [Stability of movement; fluctuations, aerodynamics]. Vol. 1. Moscow: Nauka, 1971, 255 p.

7. Kamenkov V.G. Ustoichivost' i kolebaniya nelineinykh system [Stability and oscillations of nonlinear systems]. Vol.2. Moscow: Nauka Publ., 1972, 213 p.

8. Malkin. I.G. Teoriya ustoichivosti dvizheniya [Theory of motion stability]. Moscow: Nauka Publ., 1966, 530 p.

9. Demidovich B.P. Lektsii po matematicheskoi teorii ustoichivosti [Lectures on the mathematical theory of stability]. Moscow: Nauka Publ., 1967, 472p.

10. Merkin D.R. Vvedenie v teoriyu ustoichivosti dvizheniya [Introduction to the theory of motion stability]. Moscow: Nauka Publ., 1971, 312 p.

11. Kuz'min P.A. Malye kolebaniya i ustoichivost' dvizheniya [Small oscillations and stability of motion]. Moscow: Nauka Publ., 1973, 206 p.

12. Gantmakher F.R. Teoriya matrits [Matrix Theory]. Moscow: Nauka Publ., 1967, 576 p.

13. Berezin I.S. Zhidkov N.P. Metody vychislenii [Methods of calculation]. Vol.2. Moscow: Fizmatlit Publ., 1959, 620 p.

14. Novikov M.A. Svyaz' znakoopredelennosti s privedeniem k polnym kvadratam puchka dvukh kvadratichnykh form [The connection of property of having fixed sign with reduction to a complete square of a cluster of two quadratic forms]. Vestnik Buryat. gos. un-ta. Series: Matematika i informatika  [Bulletin of Buryat State University. Series: Mathematics and Informatics], 2015, Issue 9, pp. 7–15.