Polynomial n × n matrices A(x) and B(x) over a field \( \mathbb{F} \) are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \( \mathbb{F} \) and an invertible n × n matrix Q(x) over \( \mathbb{F} \)[x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A 0x - A 1, where A 0 and A 1 are n × n matrices over \( \mathbb{F} \), and A 0 is nonsingular.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 8, pp. 1147–1152, August, 2011.
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Prokip, V.M. Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix. Ukr Math J 63, 1314–1320 (2012). https://doi.org/10.1007/s11253-012-0580-x
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DOI: https://doi.org/10.1007/s11253-012-0580-x