Field of optimal quotients and Hermitianity
Huhtanen, Marko; Kotila, Vesa (2019-02-15)
Huhtanen, M., Kotila, V. (2019) Field of optimal quotients and Hermitianity. Linear Algebra and its Applications, 563, 527-547. doi:10.1016/j.laa.2018.11.016
© 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
https://creativecommons.org/licenses/by-nc-nd/4.0/
https://urn.fi/URN:NBN:fi-fe201902195447
Tiivistelmä
Abstract
For an eigenvalue problem, be it generalized or not, the field of optimal quotients is an inclusion region containing the eigenvalues. Convexity properties, connectedness and continuity of this set are addressed. Since any notion of quotients is inherently basis dependent, varying the basis is shown to provide a lot of information. Then, due to its non-convexity, the field of optimal quotients allows recovering the spectrum exactly. In the case of a standard eigenvalue problem, the classical field of values of a matrix is recovered through a limit process. The notion leads to a serious claim how normal, Hermitian and unitary generalized eigenvalue problems should be formulated.
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