  Various algebraic multigrid algorithms have been developed for solving
problems in scientific and engineering computation over the past decades. They
have been shown to be well-suited for solving discretized partial differential
equations on unstructured girds in practice. One key ingredient of algebraic
multigrid algorithms is a strategy for constructing an effective prolongation
operator. Among many questions on constructing a prolongation, an important
question is how to evaluate its quality. In this paper, we establish new
characterizations (including sufficient condition, necessary condition, and
equivalent condition) of the so-called ideal interpolation operator. Our result
suggests that, compared with common wisdom, one has more room to construct an
ideal interpolation, which can provide new insights for designing algebraic
multigrid algorithms. Moreover, we derive a new expression for a class of ideal
interpolation operators.
