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Zbl 0657.15008
Hong, YooPyo; Horn, Roger A.
A canonical form for matrices under consimilarity.
(English)
[J] Linear Algebra Appl. 102, 143-168 (1988). ISSN 0024-3795

Complex $n\times n$ matrices A, B are said to be consimilar if $B=S\sp{- 1}A\bar S$ 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 $A\bar A$, 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) $A\bar A$ and $B\bar B$ are similar and (b) $A,A\bar A,A\bar AA,...$ have the same respective ranks as $B,B\bar B,B\bar BB,...$. Again, it is noted that A is consimilar to $\bar A,$ $A\sp{\top}$ and $A\sp*$, 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.
[G.E.Wall]
MSC 2000:
*15A21 Canonical forms, etc.
15A04 Linear transformations (linear algebra)

Keywords: semilinear transformation; consimilarity; canonical form; Jordan normal form

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