Let \begin{document}$ \mathbb{F}_{p^m} $\end{document} be a finite field of \begin{document}$ p^m $\end{document} elements, where \begin{document}$ p $\end{document} is a prime number and \begin{document}$ m $\end{document} is a positive integer. Let \begin{document}$ e\geq 2 $\end{document} be an integer and set \begin{document}$ R = \mathbb{F}_{p^m}[u]/\langle u^e\rangle $\end{document} Then \begin{document}$ R $\end{document} is a finite chain ring with unit group \begin{document}$ R^\times $\end{document} and any Type-\begin{document}$ 1 $\end{document} unit of \begin{document}$ R $\end{document} can be expressed as: \begin{document}$ \lambda+\omega u $\end{document}, where \begin{document}$ \lambda\in \mathbb{F}_{p^m}^\times $\end{document} and \begin{document}$ \omega\in R^\times $\end{document}. Let \begin{document}$ k, n $\end{document} be positive integers satisfying \begin{document}$ {\rm gcd}(p, n) = 1 $\end{document}. First, we give an explicit representation for all distinct \begin{document}$ (\lambda+\omega u) $\end{document}-constacyclic codes of length \begin{document}$ p^kn $\end{document} over \begin{document}$ R $\end{document}. Then we prove that every \begin{document}$ (\lambda+\omega u) $\end{document}-constacyclic code of length \begin{document}$ p^kn $\end{document} over \begin{document}$ R $\end{document} is monomially equivalent to a matrix-product code of a nested sequence of \begin{document}$ p^k $\end{document}\begin{document}$ \lambda_0 $\end{document}-constacyclic codes with length \begin{document}$ n $\end{document} over \begin{document}$ R $\end{document} and a \begin{document}$ p^k\times p^k $\end{document} matrix over \begin{document}$ \mathbb{F}_p $\end{document}, where \begin{document}$ \lambda_0\in \mathbb{F}_{p^m}^\times $\end{document} satisfying \begin{document}$ \lambda_0^{p^k} = \lambda $\end{document}. Using the matrix-product structures, we give an iterative construction of every \begin{document}$ (\lambda_0^{p^k}+\omega u) $\end{document}-constacyclic code of length \begin{document}$ p^kn $\end{document} by \begin{document}$ p $\end{document}\begin{document}$ (\lambda_0^{p^{k-1}}+\omega u) $\end{document}-constacyclic codes of length \begin{document}$ p^{k-1}n $\end{document} over \begin{document}$ R $\end{document}. As an application, we provide several illustrative examples for Type-\begin{document}$ 1 $\end{document} repeated-root constacyclic codes over \begin{document}$ R $\end{document}.
