A new proof of Fillmore’s theorem for integer matrices
DOI:
https://doi.org/10.22199/issn.0717-6279-5324Keywords:
inverse problem, similarity, diagonal, integer matrix, fillmoreAbstract
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.
References
A. Borobia, “Fillmore’s Theorem for integer matrices”, Linear Algebra and its Applications, vol. 531, pp. 281-284, 2017. https://doi.org/10.1016/j.laa.2017.06.011
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 Linear Algebra, vol. 35, pp 53-64, 2019. https://doi.org/10.13001/1081-3810.3886
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
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