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basis for 25.j chains of length characteristic equation cofactors column divisor column of T(z corollary element of A(z ELEMENTS ARE HOLOMORPHIC exists a matrix expressible in terms finite number finite set follows from Theorem FORM FOR MATRICES function whose expansion functions holomorphic greatest common divisor h at z0 hence holomorphic functions identically zero index range Jordan form Jordan matrix Jordan normal form jth column least common multiple linear relation exists MATRICES OF HOLOMORPHIC MATRICES WHOSE ELEMENTS matrix S(z number of terms order h partial basis point z0 exists principal ideal ring proper chains satisfies an equation satisfying the condition satisfying the equation satisfying the relations set of functions set of points set of vectors similarity transformation solution of 25.j theory of matrices transformation T(z unimodular vector chains vector equation vector whose components W. G. LEAVITT Weyr characteristic William G zero function zero of order zq exists