  For any prime number $p$, positive integers $m, k, n$ satisfying ${\rm
gcd}(p,n)=1$ and $\lambda_0\in \mathbb{F}_{p^m}^\times$, we prove that any
$\lambda_0^{p^k}$-constacyclic code of length $p^kn$ over the finite field
$\mathbb{F}_{p^m}$ is monomially equivalent to a matrix-product code of a
nested sequence of $p^k$ $\lambda_0$-constacyclic codes with length $n$ over
$\mathbb{F}_{p^m}$.
