Geodesic
On Matrices Having $J_m(1)\oplus J_m(1)$ as the Cosquare
Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 65-74

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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 $\lambda_0$.
Mots-clés : congruences, rational algorithm.
Keywords: canonical form, cosquare 
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     author = {Kh. D. Ikramov},
     title = {On {Matrices} {Having} $J_m(1)\oplus J_m(1)$ as the {Cosquare}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {65--74},
     publisher = {mathdoc},
     volume = {110},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a5/}
}
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Kh. D. Ikramov. On Matrices Having $J_m(1)\oplus J_m(1)$ as the Cosquare. Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 65-74. http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a5/