On generators of a matrix algebra and some of its subalgebras

A. A. Azamov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 1, pp. 8-14

Résumé

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 $T_n$.

Détail

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

Bibliographie

[1] Kostov V.P., “The minimal number of generators of a matrix algebra”, J. Dynamic. Control Systems, 2:4 (1996), 549–555 | DOI | MR

[2] Pirs R., Assotsiativnye algebry, Mir, M., 1986, 543 pp. | MR

[3] Laffey T.J., “Simultaneous reduction of sets of matrices under similarity”, Linear Algebra Appl., 84 (1986), 123–138 | DOI | MR

[4] Laffey T.J., “Algebras generating by two idempotetnts”, Linear Algebra Appl., 37 (1981), 45–53 | DOI | MR

[5] Rowen L., Segev Y., “Associated and Jordan algebras generated by two idempotetns”, [e-resource], 2016, 11 pp., arXiv: 1609.04899 | MR

[6] Vais I., “Algebras that are generated by two idempotents”, Seminar Analysis (Berlin, 1987/1988), Akademie-Verlag, Berlin, 1988, 139–145 | MR

[7] Aslaksen H., Sletsjøe Arne B., “Generators of matrix algebras in dimension 2 and 3”, Linear Algebra Appl., 430:1 (2009), 1–6 | DOI | MR

[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, 169, Amer. Math. Soc., Providence, 1995, 121–143 | MR

[9] Varden B.L., Algebra, Nauka, M., 1976, 648 pp. | MR

[10] Tyrtyshnikov E.E., Matrichnyi analiz i lineinaya algebra, Fizmatlit, M., 2005, 358 pp.

[11] Davis P.J., Circulant matrices, Second edition, Amer. Math. Soc., Providence, 1994, 250 pp. | MR

[12] Kirsanov Yu.A., Tsiklicheskie teplovye protsessy i teoriya teploprovodnosti v regenerativnykh vozdukhopodogrevatelyakh, Fizmatlit, M., 2007, 240 pp.

[13] Lee Chi-Liang, “Regenerative air preheaters with four channels in a power plant system”, J. Chinese Inst. Eng., 77:5 (2009), 703–710 | DOI

[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, 23:1 (2017), 12–19 | DOI | MR

[15] Azamov A.A., Bekimov M.A., “Simplified model of the heatexchange process in rotary regenerative air pre-heaters”, Ural Math. J., 2:2 (2016), 27–36 | DOI | MR

[16] Romanko V.K., Kurs raznostnykh uravnenii, Fizmatlit, M., 2012, 200 pp.