# How to find Suleimanova's work on the Nonnegative Inverse Eigenvalue Problem?

Many papers cite the work of Suleimanova when studying inverse eigenvalue problems - in particular, the nonnegative inverse eigenvalue problem (NIEP). However, I cannot seem to find her work anywhere. She proves an important result for lists of real eigenvalues. There is apparently one paper in Russian... somewhere.

Does anyone know where I can find her work?

How can people cite her work, if there's only one known paper that's in Russian?

Thanks.

• If the name ends in "ova" then it's a she. – Anthony Quas Dec 13 '16 at 6:14
• Are you referring to Suleimanova's "Stochastic matrices with real eigenvalues" (1949)? – Rodrigo de Azevedo Dec 13 '16 at 6:15
• The reference in Russian is Сулейманова X., Стохастические матрицы с вещественными собственными значениями, Докл. АН СССР. 1949. - V. 66.-Р. 343-345. – Rodrigo de Azevedo Dec 13 '16 at 6:30
• – Rodrigo de Azevedo Dec 13 '16 at 6:56
• Three things: 1) Some of the Russian journals were translated at some times. I'm not sure about that one at that time; 2) if this is a standard fact in your field, it is quite likely that some textbook will have include a proof of Suleimanova's theorem; 3) Some of the Russian journals from that era were just for statements of results, with proofs to be found elsewhere. The fact that the paper in question is a 3 page paper suggests that this is the case here. – Anthony Quas Dec 13 '16 at 7:35

Few things to note:

1. From the Math Reviews:

MR0046598 (13,760a) Reviewed Perfect, Hazel, On positive stochastic matrices with real characteristic roots. Proc. Cambridge Philos. Soc. 48, (1952). 271–276.

The author, continuing the investigations of Suleĭmanova [Doklady Akad. Nauk SSSR (N.S.) 66, 343–345 (1949); MR0030496], finds that in order that the real numbers 1,a,b, with |a|,|b|<1, shall be characteristic roots of a positive stochastic matrix of order 3 with three linearly independent characteristic vectors, it is necessary and sufficient that 1+a+b be positive. The corresponding condition for fourth order matrices is sufficient, but is shown by a counterexample not to be necessary, contrary to Suleĭmanova's assertion.

The last sentence suggests that you should be very cautious citing/using results of Suleĭmanova's papers.

1. AMS started its translations of Soviet Math. Doklady in 1960, so that is of no use for you.

2. Suleimanova published a detailed version of her Doklady paper in

Suleĭmanova, H. R. The question of a necessary and sufficient condition for the existence of a stochastic matrix with prescribed characteristic numbers. (Russian) Trudy Vsesojuz. Zaočn. Ènerget. Inst. Vyp. 28 1965 33–49.

which is an very obscure publication.

1. If you feel like you have to cite a particular result from Suleĭmanova's paper, you should first find out if this result is correct; for instance, check (using mathreviews) if this particular result was reproven later on in a paper you can trust, or prove/disprove the results yourself. If that does not work, you have no option but to get access to her papers (the interlibrary loan would help if you are in the US) and then ask somebody to translate for you.

There is a survey of the NIEP (by Egleston, Lenker and Narayan) at http://dx.doi.org/10.1016/j.laa.2003.10.019. This cites Suleĭmanova but immediately says that a simple proof of her main result was subsequently given by Perfect (http://dx.doi.org/10.1215/S0012-7094-53-02040-7). Note that this is a different paper by Perfect than the one mentioned by Misha. I presume that people decided that Suleĭmanova's work was hard to get hold of and/or not reliable, but that the results that they needed had been proved properly by Perfect. Given this, the line of least resistance is to cite both Suleĭmanova and Perfect, but in practice rely on Perfect only. That's not ideal, but it's easy to see why people do it.

• Or should we say "That's not perfect"? – Anthony Quas Dec 13 '16 at 12:51

A multi-set of real numbers $\Lambda = \{\lambda_1,\dots,\lambda_n\}$ is called a Suleĭmanova spectrum if $\Lambda$ contains one positive element and $\sum_{i=1}^n \lambda_i \geq 0$.

Suleĭmanova [Doklady Akad. Nauk SSSR (N.S.) 66, 343–345 (1949); MR0030496] loosely proved that every such set is realizable, i.e., that it is the spectrum of an $n$-by-$n$ nonnegative matrix.

Appendix B of Joanne Swift's 1972 master's thesis Location of the Characteristic Roots of Stochastic Matrices at McGill University contains an English translation of Suleĭmanova's seminal paper.

Friedland [Israel J. Math. 29, no. 1, 43–60 (1978); MR0492634] and Perfect [Duke Math. J. 20, 395–404 (1953); MR0055969] proved Suleĭmanova's result via companion matrices. However, constructing the companion matrix of a Suleĭmanova spectrum is computationally prohibitive. Recently, I gave a constructive proof of Suleĭmanova's result [Electron. J. Linear Algebra 31, 306–312 (2016); MR3504411] using permutative matrices.