We propose a recursive illumination framework for light transport analysis of an unknown scene. In this framework, we repeatedly illuminate a scene by capturing a photo and projecting the photo back onto the scene. This recursive illumination procedure produces a set of photos that spans the Krylov subspace of the light transport matrix. Many iterative methods in numerical linear algebra rely on this Krylov subspace for solving the generalized eigenvalue and minimal residual problems of large systems. Our framework expresses this recursive illumination procedure as a means to solve novel problems in computational illumination. In particular, we discuss some applications in light transport segmentation and light transport compensation. Light transport segmentation identifies pixels that mutually illuminate each other by segmenting the image according to low-level global illumination cues. Light transport compensation improves image projection quality by constructing an environment-aware projector that iteratively corrects for defocus, distortions, and global lighting. We also discuss the numerical stability of Krylov subspace illumination by considering the effects of sensor noise and signal quantization.