A new proof of Fillmore’s theorem for integer matrices

• Velasco Olalla, Rocio ^[1]
  1. [1] Universidad Nacional de Educación a Distancia

     Universidad Nacional de Educación a Distancia

     Madrid, España

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 41, Nº. 4, 2022, págs. 933-940
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5324
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• Resumen

  • Fillmore’s theorem is a matrix completion problem that states that if A is a nonscalar matrix over a field F and ϒ1,..., ϒ n ∈ F so that ϒ 1 +...+ ϒ n = tr(A) then there is a matrix similar to A with diagonal (ϒ1,..., ϒn). Borobia [1] extended Fillmore’s Theorem to the matrices over the ring of integers and Soto, Julio and Collao [3] studied it with the nonnegativity hypothesis. In this paper we prove the same result by modifying the initial proof of Fillmore, a subsequent new algorithm is proposed and some new information about the final matrix will be given.

• Referencias bibliográficas
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  • P. A. Fillmore, “On similarity and the diagonal of a matrix”, The American Mathematical Monthly, vol. 76, no. 2, pp. 167-169, 1969. https://doi.org/10.2307/2317264
  • R. L. Soto, A. I. Julio, M. A. Collao, “Brauer’s theorem and nonnegative matrices with prescribed diagonal entries”, Electronic Journal of...
  • X. Zhan, Matrix Theory. Providence, RI: American Mathematical Society, 2013.
  • Y.-J. Tan, “Fillmore’s theorem for matrices over factorial rings”, Linear and Multilinear Algebra, vol. 68, no. 3, pp. 563-567, 2018. https://doi.org/10.1080/03081087.2018.1509045