Exercise 9, § 5

Let $A$ be a square matrix of order $n$, let $B$ be a square matrix or order $m$, and let $C$ be a matrix with $m$ rows and $n$ columns over a field $K$. Reduce the equation $XA = BX + C$, where $X$ is an unknown matrix with $m$ rows and $n$ columns over $K$, to a system of linear congruences in the polynomial ring $K[Y]$ (for all $f \in K[Y]$, notice that $Xf(A) = f(B)X + H(f)$ for some well defined matrix $H(f)$ with $m$ rows and $n$ columns; let $\oplus E_i$ be the decomposition of the module $E$ associated to $A$ (Sect. 1) as a direct sum of cyclic modules, whose annihilators are the similarity invariants $f_i$ of $A$ which are distinct from 1 (Prop. 2), and let $a_i$ be generators of the $E_i$; in the same way define the submodules $F_j$, generated by $b_j$, and the polynomials $g_j$ corresponding to $B$; write $Xf_i(A)a_i = 0$; then decompose $Xu$, and $H(f_j)a_i$ in terms of the $F_j$, and show that the conditions thus obtained from the equations $Xf_i(A)a_i = 0$ are necessary and sufficient for the existence of a solution).