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

Abstract

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. TnvssKv [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. 2. Theorem 1. Any matrix S for which (1) holds can be expressed in the form S = (trace(2at~k)) where 2 is an element in the field Q(~), generated by a root ~ o f f ( x ) = 0 over the rationals, and ~1 . . . . . ~, is a characteristic vector of A in Q(~). Proof. Let ~ be a fixed root of f ( x ) = 0 and ~1 . . . . . ~, a characteristic vector of A with respect to ~. The elements ~ can be chosen in the ring Z(~), i.e. polynomials in • with integral coefficients. It is known that the ~, form the basis of an ideal a in the ring Z(~) (see [5]). Similarly, a characteristic vector ~'~, ..., ~', of A' corresponding to ~ can be chosen to form a special basis of an ideal a' in the same field, the complementary ideal to a (see [6]). This basis has the property that trace(~ia~)= 6~k. On the other hand