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Simultaneous similarity of matrices. (English) Zbl 0532.15009

In this long and important paper, the author gives a complete and explicit solution to the long-standing problem of classifying pairs of \(n\times n\) complex matrices (A,B) under simultaneous similarity \((T^{- 1}AT,T^{-1}BT)\). Let \(M_ n\) denote the set of \(n\times n\) complex matrices. The classification breaks down into several stages. At each stage an algebraic set \({\mathcal M}\subseteq M_ n\times M_ n\) is considered. Then the author constructs a finite number of rational functions \(\phi_ 1,\phi_ 2,...,\phi_ s\) in the entries of A and B, whose values are constant on all pairs similar in \({\mathcal M}\) to (A,B). The values of the functions \(\phi_ i(A,B)\), \(i=1,2,...,s\) determine a finite number of similarity classes in \({\mathcal M}\). \(S_ n\) denotes the subspaces of complex symmetric matrices in \(M_ n\). For \((A,B)\in S_ n\times S_ n\) the similarity class \((TAT^ t,TBT^ t)\) is considered, where T ranges over all complex orthogonal matrices. Then the characteristic polynomial \(\det(\lambda I-(A+xB))\) determines a finite number of similarity classes for almost all pairs \((A,B)\in S_ n\times S_ n.\)
The ideas used come mainly from algebraic geometry. To illustrate the complexity of the general problem the author gives a direct classification in the 2\(\times 2\) case, and this is highly nontrivial. He concludes the paper with some open questions and conjectures.
Reviewer: F.J.Gaines

MSC:

15A21 Canonical forms, reductions, classification
14A10 Varieties and morphisms
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