On the similarity transformation between an integral matrix with irreducible characteristic polynomial and its transpose

Abstract

Let A and S be matrices of rational integers and let A have a characteristic polynomial f which is irreducible over the rationals. Let (1) S⁻¹AS = A'. A great deal can be said about the structure of all possible S's. In first place S is symmetric by an earlier result of H. ZASSENHAUS and O. TAUSSKY [3]. In 2. an explicit expression for S will be given which is already contained in FADDEEV [2], TAUSSKY [4], BENDER [1]. In 3. the determinant of S is discussed. In 4. a fractional ideal associated with the set of all solutions of (1) is constructed.

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