Geodesic
Pairs of symmetric and skew-symmetric Toeplitz matrices whose squares coincide
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXV, Tome 514 (2022), pp. 193-203

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A description of some sets of pairs of symmetric and skew-symmetric Toeplitz matrices whose squares coincide is provided. 
@article{ZNSL_2022_514_a10,
     author = {V. N. Chugunov},
     title = {Pairs of symmetric and skew-symmetric {Toeplitz} matrices whose squares coincide},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {193--203},
     publisher = {mathdoc},
     volume = {514},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_514_a10/}
}
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V. N. Chugunov. Pairs of symmetric and skew-symmetric Toeplitz matrices whose squares coincide. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXV, Tome 514 (2022), pp. 193-203. http://geodesic.mathdoc.fr/item/ZNSL_2022_514_a10/