A Limiting Property of the Matrix Exponential

A limiting property of the matrix exponential is proven: if the (1,1)-block of a 2-by-2 block matrix becomes “arbitrarily small” in a limiting process, the matrix exponential of that matrix tends to zero in the (1,1)-, (1,2)-, and (2,1)-blocks. The limiting process is such that either the log norm of the (1,1)-block goes to negative infinity, or, for a certain polynomial dependency, the matrix associated with the largest power of the variable that tends to infinity is stable. The limiting property is useful for simplification of dynamic systems that exhibit modes with sufficiently different time scales. The obtained limit then implies the decoupling of the corresponding dynamics.