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To understand the governing mechanisms of bio-inspired swimming has always been challenging due to intense interactions between the flexible bodies of natural aquatic species and water around them. In this paper, we employ advanced modal decomposition techniques; proper orthogonal decomposition and dynamic mode decomposition, to extract energetically strongest spatio-temporal orthonormal components of complex kinematics of a Crevalle Jack (Caranx hippos) fish. Then, we present a computational framework for handling fluid-structure interaction related problems in order to investigate their contributions towards the overall dynamics of highly nonlinear systems. We find that the undulating motion of this fish can be described by only two standing-wave like spatially orthonormal modes. Constructing the data set from our numerical simulations for flows over the membranous caudal fin of the Jack fish, our modal analyses reveal that only the first few modes receive energy from both the fluid and structure, but the contribution of fluid in the higher modes is minimal. For the viscous and transitional flow conditions considered here, both spatially and temporally orthonormal modes show strikingly similar coherent flow structures. Our investigations are expected to assist in developing data-driven reduced-dimensional mathematical models to examine the dynamics of bio-inspired swimming robots and develop new and effective control strategies to bring their performance closer to real fish species.