Plasmonic surface lattice resonances (SLRs), characterized by low radiative loss, play an important role in nanophotonics and optical metasurfaces. However, conventional coupled dipole approximation (CDA) treatments are restricted to deterministic homogeneous excitation, resulting in limited flexibility and restricted degrees of freedom. Here, by combining CDA with singular-value decomposition, we establish a unified theoretical framework that reformulates SLRs as a physical eigenproblem connecting two distinct Hilbert spaces. This approach yields orthogonal basis sets for arbitrary lattice geometries, and provides a general description of plasmonic lattice response under arbitrary excitation, thereby naturally enabling the joint optimization of incident fields and structural geometries. This work provides deeper insight into light-lattice interactions and paves the way for designing advanced SLR-based applications.