Carbon fiber reinforced polymer (CFRP) is widely utilized due to their superior properties. The structural health monitoring (SHM) for composite materials is crucial to prevent catastrophic failures. Electrical impedance tomography (EIT), which allows for the visualization of the conductivity distribution through the measurement of electrical parameters on the boundary, presents a potential solution for SHM. Existing studies applying EIT to CFRP concentrate on traditional linearization methods that rely on sensitivity matrix, which ignore challenges posed by non-uniform sensitivity. This paper introduces an EIT image reconstruction method that is independent of the sensitivity matrix, offering a novel way for rapid damage detection and real-time SHM. Furthermore, the domain shape transformation is achieved by conformal mapping. Simulations and experiments indicate that the sensitivity-matrix-independent EIT can avoid issues such as blurring in the central region and artifacts near the boundary caused by non-uniform distribution of the sensitivity in the CFRP damage detection.