The determination of sets of conjugates to a given matrix M with respect to its minimum equation was first discussed by E. S. Sokolnikoff in 1933.f In two papers published more recently, one J by E. T. Browne and one § by the writer, the subject has received further consideration, and, in particular, attention has been paid to the case in which M has only linear elementary divisors. In the former of these two papers, conjugates were obtained by employing the principal idempotent elements of M. The second paper considered only conjugates of M in a ring R{A) where A had simple latent roots, and made use of the principal idempotent elements of A, Since each principal idempotent element of If is a set of one or more of its partial idempotent elements, there will be, in general, more sets of conjugates obtained by using the latter than by using the former. This fact was stated by both Sokolnikoff and Browne. Not all of these conjugates are polynomials in M. The writer exhibited sets of matrices in the ring R(A) conjugate to M but not expressible as polynomials in M. These are accounted for by the fact that the principal idempotent elements of A are the partial idempotent elements of M. It is the purpose of this paper to show that there exists a nonderogatory matrix A (by no means unique), such that any matrix M can be expressed as a polynomial in A. Then for the restricted case under consideration, the exact numbers of conjugate sets in the rings R(A) and R(M) can be determined.
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