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Certain Properties of Square Matrices over Fields with Applications to Rings
Algunas propiedades de matrices cuadradas sobre cuerpos con aplicaciones a anillos
DOI:
https://doi.org/10.15446/recolma.v54n2.93833Palabras clave:
Nilpotent matrices, idempotent matrices, Jordan canonical form, algebraically closed fields, super pi-regular rings (en)Matrices nilpotentes, matrices idempotentes, forma canónica de Jordan, cuerpos algebraicamente cerrados, anillos pi-regulares (es)
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We prove that any square nilpotent matrix over a field is a difference of two idempotent matrices as well as that any square matrix over an algebraically closed field is a sum of a nilpotent square-zero matrix and a diagonalizable matrix. We further apply these two assertions to a variation of π-regular rings. These results somewhat improve on establishments due to Breaz from Linear Algebra & amp; Appl. (2018) and Abyzov from Siberian Math. J. (2019) as well as they also refine two recent achievements due to the present author, published in Vest. St. Petersburg Univ. - Ser. Math., Mech. & amp; Astr. (2019) and Chebyshevskii Sb. (2019), respectively.
Probamos que toda matriz cuadrada nilpotente sobre un cuerpo es igual a la resta de dos matrices idempotentes, también probamos que toda matriz cuadrada con coeficientes en un cuerpo algebraicamente cerrado es la suma de una matriz nilpotente cuyo cuadrado es nulo y una matriz diagonalizable. También aplicamos estos resultados en una variante de anillos π-regulares. Estos resultados mejoran los resultados presentados por Breaz en Linear Algebra & Appl. (2018) y aquellos de Abyzov presentados en Siberian Math. J. (2019) al igual que aquellos publicados por el autor del presente artículo en Vest. St. Petersburg Univ. - Ser. Math., Mech. & Astr. (2019) y en Chebyshevskii Sb. (2019), respectivamente.
Referencias
A. N. Abyzov, Strongly q-nil-clean rings, Siber. Math. J. 60 (2019), no. 2, 197-208. DOI: https://doi.org/10.1134/S0037446619020022
A. N. Abyzov and I. I. Mukhametgaliev, On some matrix analogs of the little Fermat theorem, Math. Notes 101 (2017), no. 1-2, 187-192. DOI: https://doi.org/10.1134/S0001434617010229
K. I. Beidar, K. C. O'Meara, and R. M. Raphael, On uniform diagonalisation of matrices over regular rings and one-accesible regular algebras, Commun. Algebra 32 (2004), 3543-3562. DOI: https://doi.org/10.1081/AGB-120039630
W. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations, Designs, Codes and Cryptography, Springer Verlag, 70, 2014, 405-431. DOI: https://doi.org/10.1007/s10623-012-9704-4
S. Breaz, Matrices over finite fields as sums of periodic and nilpotent elements, Lin. Alg. & Appl. 555 (2018), 92-97. DOI: https://doi.org/10.1016/j.laa.2018.06.017
S. Breaz, G. Cälugäreanu, P. Danchev, and T. Micu, Nil-clean matrix rings, Lin. Alg. & Appl. 439 (2013), 3115-3119. DOI: https://doi.org/10.1016/j.laa.2013.08.027
M. P. Cuéllar, J. Gómez-Torrecillas, F. J. Lobillo, and G. G. Navarro, Genetic algorithms with permutation-based representation for computing the distance of linear codes, arXiv:2002.12330v1, arXiv:1810.01260.
P. Danchev, E. García, and M. G. Lozano, Decompositions of matrices into diagonalizable and square-zero matrices, Lin. & Multilin. Algebra 69 (2021). DOI: https://doi.org/10.1080/03081087.2020.1862742
P. V. Danchev, A generalization of pi-regular rings, Turk. J. Math. 43 (2019), 702-711. DOI: https://doi.org/10.3906/mat-1808-74
P. V. Danchev, On a property of nilpotent matrices over an algebraically closed field, Chebyshevskii Sbornik 20 (2019), no. 3, 400-403. DOI: https://doi.org/10.22405/2226-8383-2019-20-3-401-404
P. V. Danchev, Weakly exchange rings whose units are sums of two idempotents, Vestnik of St. Petersburg Univ., Ser. Math., Mech. & Astr. 6(64) (2019), no. 2, 265-269. DOI: https://doi.org/10.21638/11701/spbu01.2019.208
P. V. Danchev, Representing matrices over fields as square-zero matrices and diagonal matrices, Chebyshevskii Sbornik 21 (2020), no. 3. DOI: https://doi.org/10.22405/2226-8383-2020-21-3-84-88
E. García, M. G. Lozano, R. M. Alcázar, and G. Vera de Salas, A Jordan canonical form for nilpotent elements in an arbitrary ring, Lin. Alg. & Appl. 581 (2019), 324-335. DOI: https://doi.org/10.1016/j.laa.2019.07.016
H. Gluesing-Luerssen, Introduction to skew-polynomial rings and skew-cyclic codes, arXiv:1902.03516v2.
J. Gómez-Torrecillas, F. J. Lobillo, and G. Navarro, Convolutional codes with a matrix-algebra wordambient, Advances in Mathematics of Communications 10 (2016), 29-43. DOI: https://doi.org/10.3934/amc.2016.10.29
J. Gómez-Torrecillas, F. J. Lobillo, and G. Navarro, A new perspective of cyclicity in convolutional codes, IEEE Transactions on Information Theory 62 (2016), 2702-2706. DOI: https://doi.org/10.1109/TIT.2016.2538264
J. Gómez-Torrecillas, F. J. Lobillo, and G. Navarro, Ideal codes over separable ring extensions, IEEE Transactions on Information Theory 63 (2017), 2796-2813. DOI: https://doi.org/10.1109/TIT.2017.2731774
R. E. Hartwig and M. S. Putcha, When is a matrix a difference of two idempotents, Lin. & Multilin. Algebra 26 (1990), no. 4, 267-277. DOI: https://doi.org/10.1080/03081089008817983
D. A. Jaume and R. Sota, On the core-nilpotent decomposition of trees, Lin. Alg. & Appl. 563 (2019), 207-214. DOI: https://doi.org/10.1016/j.laa.2018.10.012
O. Lezama, Coding theory over noncommutative rings of polynomial type, preprint (2020).
K. C. O'Meara, Nilpotents often the difference of two idempotents, unpublished draft privately circulated on March 2018.
Y. Shitov, The ring M8k+4(Z2) is nil-clean of index four, Indag. Math. 30 (2019), 1077-1078. DOI: https://doi.org/10.1016/j.indag.2019.08.002
J. Ster, On expressing matrices over Z2 as the sum of an idempotent and a nilpotent, Lin. Alg. & Appl. 544 (2018), 339-349. DOI: https://doi.org/10.1016/j.laa.2018.01.015
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1. P.V. Danchev. (2021). On the idempotent and nilpotent sum numbers of matrices over certain indecomposable rings and related concepts. Matematychni Studii, 55(1), p.24. https://doi.org/10.30970/ms.55.1.24-32.
