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The paper presents an algebra-geometric approach for determining the time-domain reduced-order models and frequency-domain reduced-degree models of large-scale multivariable systems. First, the structures of the canonical state-space representations and corresponding matrix fraction descriptions of general multivariable systems are introduced, and the associated characteristic Â¿-matrices are defined. Next, the divisors and spectral decomposition theorems for the nonsingular characteristic Â¿-matrices, which may not be regular or monic, are developed by using the algebraic and geometric properties of multivariable-system structures. Then, the derived algebra-geometric theorems are used to develop a frequency-domain aggregation method and a time-domain aggregation method for the model reduction of large-scale multivariable systems. Finally, the newly developed matrix sign functions, in conjunction with the aggregation method, are used to obtain the reduced-order and reduced-degree models of large-scale multivariable systems, without assuming that the eigenvalues of the original systems are known and/or the singularly-perturbed models are available.