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In this paper, algorithms are presented for obtaining relatively prime matrix fraction descriptions (MFD's) of linear time-invariant systems. Orthogonal coordinate transformations are used to reduce a given state-space model to one whose state matrix is in "block Hessenberg" form. The observability (controllability) indexes of the system are also obtained in the process. A recursive algorithm is then described for obtaining a relatively prime MFD from the block Hessenberg representation. It is then shown that some simplification can be made in the recursive algorithm by first reducing the block Hessenberg form, by means of a nonsingular (triangular) coordinate transformation, to a more compact form, such as the "block Frobenius" form. A permutation of the state variables of the block Frobenius form yields a canonical representation similar to the Luenberger canonical form. Some numerical and other properties of the algorithms are discussed, and the use of the algorithms is illustrated by numerical examples. The numerical performance of the algorithms is also compared with that of the structure algorithm of Wolovich and Falb.