  For each pair of matrices $A$ and $B$ with the same order, let $\|A-B\|_F$
denote their Frobenius distance. This paper deals mainly with the Frobenius
distances from projections to an idempotent matrix. For every idempotent $Q\in
\mathbb{C}^{n\times n}$, a projection $m(Q)$ called the matched projection can
be induced. It is proved that $m(Q)$ is the unique projection whose Frobenius
distance away from $Q$ takes the minimum value among all the Frobenius
distances from projections to $Q$, while $I_n-m(Q)$ is the unique projection
whose Frobenius distance away from $Q$ takes the maximum value. Furthermore, it
is proved that for every number $\alpha$ between the minimum value and the
maximum value, there exists a projection $P$ whose Frobenius distance away from
$Q$ takes the value $\alpha$. Based on the above characterization of the
minimum distance, some Frobenius norm upper bounds and lower bounds of
$\|P-Q\|_F$ are derived under the condition of $PQ=Q$ on a projection $P$ and
an idempotent $Q$.
