On generators of a matrix algebra and some of its subalgebras

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 $T_n$.