Abstract
The nonnegative inverse eigenvalue problem (NIEP) asks which lists of n complex numbers (counting multiplicity) occur as the eigenvalues of some n-by-n entry-wise nonnegative matrix. The NIEP has a long history and is a known hard (perhaps the hardest in matrix analysis?) and sought after problem. Thus, there are many subproblems and relevant results in a variety of directions. We survey most work on the problem and its several variants, with an emphasis on recent results, and include 130 references. The survey is divided into: a) the single eigenvalue problems; b) necessary conditions; c) low-dimensional results; d) sufficient conditions; e) appending 0’s to achieve realizability; f) the graph NIEP’s; g) Perron similarities; and h) the relevance of Jordan structure.
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Johnson, C.R., Marijuán, C., Paparella, P., Pisonero, M. (2018). The NIEP. In: André, C., Bastos, M., Karlovich, A., Silbermann, B., Zaballa, I. (eds) Operator Theory, Operator Algebras, and Matrix Theory. Operator Theory: Advances and Applications, vol 267. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-72449-2_10
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DOI: https://doi.org/10.1007/978-3-319-72449-2_10
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-72448-5
Online ISBN: 978-3-319-72449-2
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