matrices A, B are said to be consimilar if
for some non-singular complex matrix S. This concept arises naturally from comparing the expressions for a semilinear transformation on an n-dimensional complex vector space in two different coordinate systems. The present paper gives a detailed survey, with an extensive bibliography, of the known results on consimilarity. A canonical form for A under consimilarity, closely related to the usual Jordan normal form for
, is described in section 3. Various applications are given in section 4. For example (Theorem 4.5), A and B are consimilar if, and only if, (a)
are similar and (b)
have the same respective ranks as
. Again, it is noted that A is consimilar to
, and that every matrix is consimilar both to a real matrix and to a Hermitian matrix. Altogether, this is a useful and clearly written survey.