Abstract.A permutative matrix is a square matrix such that every row is a permutation of the first row. A constructive version of a result attributed to Suleȋmanova is given via permutative matrices. A well-known result is strenghthened by showing that all realizable spectra containing at most four elements can be realized by a permutative matrix or by a direct sum of permutative matrices. The paper concludes by posing a problem.

Call an n-by-n invertible matrix S a Perron similarity if there is a real nonscalar diagonal matrix D such that SDS −1 is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra C (S) :which we call the Perron spectracone and Perron spectratope, respectively. The set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes, so that enumerating them is of interest.The Perron spectracone and spectratope of Hadamard matrices are of particular interest and tend to have large volume. For the canonical Hadamard matrix (as well as other matrices), the Perron spectratope coincides with the convex hull of its rows.In addition, we provide a constructive version of a result due to Fiedler ([9, Theorem 2.4]) for Hadamard orders, and a constructive version of [2, Theorem 5.1] for Suleȋmanova spectra.

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The question of the exact region in the complex plane of the possible single eigenvalues of all n-by-n stochastic matrices was raised by Kolmogorov in 1937 and settled by Karpelevič in 1951 after a partial result by Dmitriev and Dynkin in 1946. The Karpelevič result is unwieldy, but a simplification was given by Ðoković in 1990 and Ito in 1997. The Karpelevič region is determined by a set of boundary arcs each connecting consecutive roots of unity of order less than n. It is shown here that each of these arcs is realized by a single, somewhat simple, parameterized stochastic matrix. Other observations are made about the nature of the arcs and several further questions are raised. The doubly stochastic analog of the Karpelevič region remains open, but a conjecture about it is amplified.

Eventually positive matrices are real matrices whose powers become and remain strictly positive. As such, eventually positive matrices are a fortiori matrix roots of positive matrices, which motivates us to study the matrix roots of primitive matrices. Using classical matrix function theory and Perron-Frobenius theory, we characterize, classify, and describe in terms of the real Jordan canonical form the pth-roots of eventually positive matrices.