Abstract
If the cosquares of complex matrices \(A\) and \(B\) are similar and there is a unimodular number in the spectrum of the cosquares, then \(A\) and \(B\) are not necessarily congruent. Assume that such an eigenvalue \(\lambda_0\) is unique. In this case, so far, one could verify the congruence of \(A\) and \(B\) by using a rational algorithm only in two situations: (1) the eigenvalue \(\lambda_0\) is simple or semi-simple; (2)there is only one Jordan block associated with \(\lambda_0\) in the Jordan form of the cosquares. We propose a rational algorithm for checking congruence in the case where two Jordan blocks of the same order are associated with \(\lambda_0\).
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Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 65-74 https://doi.org/10.4213/mzm12606.
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Ikramov, K.D. On Matrices Having \(J_m(1)\oplus J_m(1)\) as Their Cosquare. Math Notes 110, 72–79 (2021). https://doi.org/10.1134/S0001434621070075
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DOI: https://doi.org/10.1134/S0001434621070075