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- W2025986492 abstract "In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT−1, TBT−1). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set M0n,2,r,π ⊆ Mn × Mn (Mn = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø1,…,øs in the entries of A and B whose values are constant on all pairs similar in Mn,2,r,π to (A, B). The values of the functions øi(A, B), i = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in Mn,2,r,π. Let Sn be the subspace of complex symmetric matrices in Mn. For (A, B) ϵSn × Sn we consider the similarity class (TATt, TBTt), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ϵSn × Sn." @default.
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- W2025986492 date "1983-12-01" @default.
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- W2025986492 title "Simultaneous similarity of matrices" @default.
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- W2025986492 doi "https://doi.org/10.1016/0001-8708(83)90044-0" @default.
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