# A. A. Azamov. On generators of a matrix algebra and some of its subalgebras... P. 8-14

It is shown that a full matrix algebra $M_n$ admits a generator system consisting of two nilpotent matrices $P$ and $Q$ such that any matrix $A=(a_{ij})$ is expressed explicitly in terms of $P$ and $Q$ as $A=\sum_{i\neq j}a_{ij}P^{i-1}QP^{n-j}$, $i,j=1,2,\ldots,n$. We show how this representation can be applied to calculate the powers of the coefficient matrix $A$ of a linear system $x_{n+1}=Ax_n+r_n$ modeling heat exchange in a regenerative air preheater. More exactly, we obtain convenient recursive formulas for the elements of $A^{k}$, $k=1,2,\ldots$. We also consider the problem of constructing a simple system of generators for the subalgebras of diagonal and triangular matrices. We observe that a generating matrix of the subalgebra of diagonal matrices is related to the Lagrange interpolation formula and prove that the subalgebra of triangular matrices is generated by a diagonal matrix with pairwise different elements and first skew diagonal. It is shown that a triangular matrix $A \in T_n$ with pairwise different diagonal elements can be reduced to a Jordan form within the subalgebra $T_n$; i.e., there exists $L\in T_n$ such that $L^{-1}AL$ is diagonal. In the general case this property does not hold for arbitrary matrices from $T_n$.

Keywords: matrix algebra, system of generators, nilpotent matrix, matrix unit, subalgebra, Jordan form, interpolation polynomial, discrete system, air preheater, heat exchange.

The paper was received by the Editorial Office on October 18, 2017.

Funding Agency:  Committee for coordination science and technology development under the Cabinet of Ministers of Uzbekistan (project no. ОТ-Ф4-84).

Abdulla Azamovich Azamov, Dr. Phys.-Math. Sci., Prof., Uzbekistan Academy of Sciences V.I.Romanovskiy Institute of Mathematics, Tashkent, 100041 Uzbekistan,
e-mail: abdulla.azamov@gmail.com.

REFERENCES

1. Kostov V.P. The minimal number of generators of a matrix algebra. J. Dynamic. Control Systems, 1996, vol. 2, no. 4, pp. 549–555. doi:  https://doi.org/10.1007/BF02254702 .

2. Pierce R.S. Associative algebras. N Y, Springer-Verlag, 1982, 436 p. doi:  https://doi.org/10.1007/978-1-4757-0163-0 . Translated to Russian under the title Assotsiativnye algebry, Moscow, Mir Publ., 1986, 543 p.

3. Laffey Thomas J. Simultaneous reduction of sets of matrices under similarity. Linear Algebra Appl., 1986, vol. 84, pp. 123–138. doi:  10.1016/0024-3795(86)90311-3 .

4. Laffey Thomas J. Algebras generating by two idempotetnts. Linear Algebra Appl., 1981, vol. 37, pp. 45–53. doi:  https: 10.1016/0024-3795(81)90166-X.

5. Rowen L., Segev Y. Associated and Jordan algebras generated by two idempotetns [e-resource]. 2016. Available at: https://arxiv.org/abs/1609.04899 . 11 p.

6. Vais I. Algebras that are generated by two idempotents. Seminar Analysis (Berlin, 1987/1988). Berlin: Akademie-Verlag, 1988, pp. 139–145.

7. Aslaksen H., Sletsjøe Arne B. Generators of matrix algebras in dimension 2 and 3. Linear Algebra Appl., 2009, vol. 430, no. 1, pp. 1–6. doi:  https://doi.org/10.1016/j.laa.2006.05.022 .

8. Popov V.L. An analogue of M.Artin’s conjecture on invariants for nonassociative algebras. Lie Groups and Lie Algebras: E.B.Dynkin’s Seminar, American Math. Soc. Trans. Ser. 2, vol. 169, Providence: Amer. Math. Soc., 1995, pp. 121–143.

9. van der Waerden B.L. Algebra I, II. Berlin, Heidelberg, Springer-Verlag, 1971, 272 p. ISBN: 3540035613 , 1967, 300 p. Translated to Russian under the title van der Varden B.L. Algebra, Moscow, Nauka Publ., 1976, 648 p.

10. Tyrtyshnikov E.E. Matrichnyi analiz i lineinaya algebra. [Matrix analysis and linear algebra]. Moscow, Fizmatlit Publ., 2005, 358 p.

11. Davis Philip J. Circulant Matrices: Second edition. Providence: American Math. Soc., 1994, 250 p. ISBN: 0828403384 .

12. Kirsanov Yu.A. Tsiklicheskie teplovye protsessy i teoriya teploprovodnosti v regenerativnykh vozdukhopodogrevatelyakh. [Cyclic thermal processes and the theory of thermal conductivity in regenerative air heaters]. Moscow, Fizmatlit Publ., 2007, 240 p. ISBN: 978-5-9221-0831-7 .

13. Lee Chi-Liang Regenerative air preheaters with four channels in a power plant system. J. Chinese Inst. Eng., 2009, vol. 77, no. 5, pp. 703–710. doi:  https://doi.org/10.1080/02533839.2009.9671552 .

14. Azamov A.A., Bekimov M.A. A discrete model of the heat exchange process in rotating regenerative air preheaters. Tr. Inst. Mat. Mekh. UrO RAN, 2017, vol. 23, no. 1, pp. 12–19. (in Russian)  doi:  https://doi.org/10.21538/0134-4889-2017-23-1-12-19 .

15. Azamov A.A., Bekimov M.A. Simplified model of the heatexchange process in rotary regenerative air pre-heaters. Ural Math. J., 2016, vol. 2, no. 2 pp. 27–36. doi:  https://doi.org/10.15826/umj.2016.2.003 .

16. Romanko V.K. Kurs raznostnykh uravnenii. [Course of difference equations]. Moscow, Fizmatlit Publ., 2012, 200 p. ISBN: 978-5-9221-1387-8 .