# Update rank-revealing factorization after applying $I \otimes$(small matrix)

Suppose one knows an $$(m, n)$$ matrix $$A$$ along with some rank-revealing factorization (let's say the economy SVD for conreteness), $$A = U S V^H$$, with $$S$$ an $$(r, r)$$ matrix where $$r$$ is the numerical rank of $$A$$. Various algorithms exist to efficiently obtain the decomposition of the low-rank update $$A + B C^H$$, where $$B$$ and $$C$$ have small rank.

Suppose one seeks an update which is not low rank, but only because $$B$$ and $$C$$ take the form e.g. $$B = I \otimes b$$, where $$b$$ is a $$(d, d)$$ matrix with $$d <<$$ any of $$r, m, n$$, and $$I$$ is the identity of dimension $$m-d$$. Can anything be said about the update in this case?