The Frobenious distances from projections to an idempotent matrix

Abstract

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 $Q$.