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.

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.

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