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I've been studying IEPs, in particular, the Nonnegative Inverse Eigenvalue Problem, some basic theoretical framework, the many open questions that IEPs have, and now sort of realize the computational difficulty of reconstructing matrices from prescribed spectral data (e.g., prescribed spectrum).

Are these problems sort of dead right now, like hopeless? Or perhaps IEPs is actively being researched today?

Most of the literature that I have found on IEPs with the most partial progress seems to be very old - late 70s to mid 80s. (Newer papers seem to just reiterate old results.)

For the nonnegative inverse eigenvalue problem, the problem is open for $n\ge 5$.

Thanks in advance for any advice or suggestions.

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  • $\begingroup$ I know it's not the same, but there is lots of current interest in matrix completion, which fits the general idea of reconstructing matrices from prescribed data. I wonder if any of that literature may be useful. $\endgroup$
    – user62562
    Commented Oct 9, 2016 at 13:19
  • $\begingroup$ These problems are interesting if you are trying to design matrices for testing out some simulations that rely on matrices with certain spectral properties. However, as far as I am aware, there are hardly any uses of IEPs (haven't looked in a long while to see if things have changed). Unless this is a hobby for you, I'd suggest not sinking too much time into these (the math is interesting, but as a research direction, of somewhat unclear significance). $\endgroup$
    – Suvrit
    Commented Oct 9, 2016 at 14:27
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    $\begingroup$ Not wishing to brag, but if you are permitted to add as many zeros as you like to the spectrum (viewed as a set with multiplicities), the problem of realizing primitive matrices with designated spectrum (modulo the multiplicity of zero) is solved: Mike Boyle and David Handelman, The Spectra of Nonnegative Matrices Via Symbolic Dynamics, Annals of Mathematics Second Series, Vol. 133, No. 2 (Mar., 1991), pp. 249-316. $\endgroup$
    – David Handelman
    Commented Oct 9, 2016 at 15:10
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    $\begingroup$ Wow, congratulations, Professor Handelman! I just found your paper and will be sure to read it over, if the level of mathematics is accessible enough to me at this point in time. Does your result mean that the NIEP is, in fact, solved for generalized $n$? Or is the problem still open for $n \ge 5$? Most of the literature that I have found so far say this, so I was wondering what you thought. Thanks so much, @DavidHandelman, $\endgroup$
    – User001
    Commented Oct 11, 2016 at 16:11
  • $\begingroup$ No; notice the restriction that the multiplicity of zero (and thus the matrix size) is uncontrolled. Thus if you have a set $S$ of $m$ nonzero complex numbers (with multiplicities) and the (necessary and sufficient) trace-like conditions of the reference are satisfied, then there exists a realization as a primitive matrix of $S \cup \brcs{0,0,0,\dots}$ (for some multiplicity of zeros, say $k$, which is very difficult to determine) as the spectrum (with multiplicities) of a primitive matrix of size $m+k$. ...ctd $\endgroup$
    – David Handelman
    Commented Oct 11, 2016 at 23:37

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Although I work exclusively on the nonnegative inverse eigenvalue problem (NIEP), I can assure you that IEPs are far from dead, both on the theoretical side and the applied side (there has been a flurry of activity in recent years, along with a special session at the 2017 Meeting of the International Linear Algebra Society).

I would like to point out that in the paper by Charlie Johnson and I [MR3452738; *Linear Algebra Appl.* 493 (2016), 281–300], connections between the real NIEP to Bose-Mesner algebras and association schemes were established.

For more details on current developments on the NIEP, see the recent survey paper (which will appear later this year in Operator Theory: Advances and Applications by Johnson et al (it is worth noting that this survey already dated needs updating).

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