The Frobenius distances from projections to an idempotent matrix.

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 ∈ C n × 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 α between the minimum value and the maximum value, there exists a projection P whose Frobenius distance away from Q takes the value α. 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 P Q = Q on a projection P and an idempotent Q. [ABSTRACT FROM AUTHOR]