Speaker
Description
The Real Nonnegative Inverse Eigenvalue problem (hereforth, RNIEP) consists, for a given positive integer n, in characterizing the lists of n real numbers which are the spectrum of some n × n matrix with real entries. C-realizability was originally introduced in [1] as a sufficient condition for the RNIEP. It was shown back then that C-realizability included as particular cases most of the known sufficient conditions for the RNIEP. It was not until 2017 that it was shown in [2] that C-realizability was more closely related to the Symmetric Nonnegative Inverse Eigenvalue problem (SNIEP)) than to the RNIEP. The combinatorial nature of the original definition of C-realizability has conduced over the years to combinatorial characterizations of the set of C-realizable lists. In this talk we first review a partial one (see [4]) for real lists with zero sum, and then the most general combinatorial characterization, obtained in [5] for arbitrary real lists. One of the most remarkable consequences of the latter is that it proves the monotonicity of C-realizability, i.e., that the operation of increasing any positive entry of a C-realizable list preserves C-realizability.
References
[1] A. Borobia, J. Moro & R. L. Soto (2008). A unified view on compensation criteria in the real nonnegative inverse eigenvalue problem. Linear Algebra Appl., 428, 2574-2584.
[2] R. Ellard & H. Smigoc (2017). Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalue Problem. Linear Algebra Appl, 498, 521-552.
[3] C. R. Johnson, T. J. Laffey & R. Loewy (1996). The Real and the Symmetric Nonnegative Inverse Eigenvalue Problem are different. Proc. Am. Math. Soc., 124, 3647-3651.
[4] C. Marijuán & J. Moro. (2021) A characterization of trace-zero sets realizable by compensation in the SNIEP. Linear Algebra Appl., 615, 42-76.
[5] C. Marijuán & J. Moro. (2024) A characterization of sets realizable by compensation in the SNIEP. Linear Algebra Appl., 693, 425–447.