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Matrix fixed-point iterations z n+1 = ψ(zn) defined by a rational function ψ are considered. For these iterations a new proof is given that matrix convergence is essentially reduced to scalar convergence. It is shown that the principal Padé family of iterations for the matrix sign function and the matrix square root is a special case of a family of rational iterations due to Ernst Schröder. This characterization provides a family of iterations for the matrix pth root which preserve thedoi:10.1137/070694351 fatcat:gj5xbv4q2nbc5h7yujyzcfz5je