  Let $e \geq 2$ be an integer, $p^r$ be a prime power with $p^r \equiv 1\
({\rm mod}\ e)$ and $\eta_r(i)$ be Gaussian periods of degree $e$ for ${\mathbb
F}_{p^r}$. By the dual form of Davenport and Hasse's lifting theorem on Gauss
sums, we establish lifts of the multiplication matrices of the Gaussian periods
$\eta_r(0),\ldots,\eta_r(e-1)$ which are defined by F. Thaine. We also give
some examples of the explicit lifts for prime degree $e$ with $3\leq e\leq 23$
which also illustrate relations among lifts of Jacobi sums, Gaussian periods
and multiplication matrices of Gaussian periods.
