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Linear optics aim at realizing any real- and/or complex-valued matrix operator via optical elements, addressing a broad field of applications in the areas of quantum photonics, microwave photonics and optical neural networks. The transfer of linear operators into photonic experimental layouts typically relies on Singular Value Decomposition (SVD) techniques combining meshes of cascaded 2x2 Mach Zehnder Interferometers (MZIs), with the main challenges being the precision in the experimental representation of the targeted matrix, referred to as fidelity, and the overall insertion loss. We demonstrate a novel interferometric coherent photonic crossbar architecture (Xbar) that demarcates from state-of-the-art SVD implementations and can realize any linear operator, supporting full restoration of the loss-induced fidelity. Its novel interferometric design allows for the direct mapping of each matrix element to a single, designated Xbar node, bringing down the number of programming steps to only one. We present the theoretical foundations of the Xbar, proving that its insertion losses scale linearly with the node losses as opposed to the exponential scaling witnessed by the SVD counterparts. This leads to a matrix design with significantly lower overall insertion losses compared to SVD-based schemes when utilizing state-of-the-art silicon photonic fabrication metrics, allowing for alternative node technologies with lower energy consumption and higher operational speed credentials to be employed. Finally, we validate that our Xbar architecture is the first linear operator that supports fidelity restoration, outperforming SVD schemes in loss- and phase-error fidelity performance and forming a significantly more robust layout to loss and phase deviations.