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A canonical form for matrices under consimilarity. (English) Zbl 0657.15008
Complex $n×n$ matrices A, B are said to be consimilar if $B={S}^{-1}A\overline{S}$ 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 $A\overline{A}$, 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) $A\overline{A}$ and $B\overline{B}$ are similar and (b) $A,A\overline{A},A\overline{A}A,···$ have the same respective ranks as $B,B\overline{B},B\overline{B}B,···$. Again, it is noted that A is consimilar to $\overline{A},$ ${A}^{\top }$ and ${A}^{*}$, 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.
Reviewer: G.E.Wall
##### MSC:
 15A21 Canonical forms, reductions, classification 15A04 Linear transformations, semilinear transformations (linear algebra)