Perturbation analysis for the matrix least squares problem AXB=CAXB=C

Let SS and Ŝ be two sets of solutions to matrix least squares problem (LSP) AXB=CAXB=C and the perturbed matrix LSP ÂX̂B̂=Ĉ, respectively, where Â=A+ΔA, B̂=B+ΔB, Ĉ=C+ΔC, and ΔAΔA, ΔBΔB, ΔCΔC are all small perturbation matrices. For any given X∈SX∈S, we deduce general formulas of the least squares solutions X̂∈Ŝ that are closest to XX under appropriated norms, meanwhile, we present the corresponding distances between them. With the obtained results, we derive perturbation bounds for the nearest least squares solutions. At last, a numerical example is provided to verify our analysis.