Suleĭmanova Spectra & Realization Theorems

In the spectral theory of nonnegative matrices, the most thoroughly understood family of spectra consists of real multisets possessing exactly one positive eigenvalue (the Perron root), with all remaining eigenvalues being nonpositive.

For general candidate spectra, spectral nonnegativity requires verifying infinitely many nonlinear power-sum inequalities ($s_k = \operatorname{Tr}(A^k) \ge 0$), Newton-Girard relations, and Johnson-Loewy-London constraints. However, when all secondary eigenvalues are nonpositive:

• Even powers ($k = 2m$): Every term satisfies $\lambda_i^{2m} \ge 0$, ensuring $s_{2m} = \sum_{i=0}^{n-1} \lambda_i^{2m} > 0$ holds unconditionally.
• Odd powers ($k = 2m+1$): Because $\lambda_0 \ge |\lambda_j|$ for all $j \ge 1$, we have $\lambda_0^{2m+1} \ge \sum_{j=1}^{n-1} |\lambda_j|^{2m+1}$ whenever $\lambda_0 \ge \sum_{j=1}^{n-1} |\lambda_j|$. Hence all odd power sums $s_{2m+1} \ge 0$ are satisfied simultaneously.

Consequently, all higher-order trace obstructions vanish identically. The linear condition $\operatorname{Tr}(A) = s_1 \ge 0$ is the sole necessary obstacle.

In 1949, the Soviet mathematician H. R. Suleĭmanova (1949) published an announcement in Doklady Akademii Nauk SSSR asserting that the trace condition $s_1 \ge 0$ is sufficient to realize any such spectrum by an entrywise stochastic matrix. However, her original paper contained only a brief sketch of an induction argument without complete proofs.

In 1953 and 1955, the British mathematician Hazel Perfect (1953, 1955) delivered the first complete, rigorous proofs using similarity transformations on companion matrices:

Perfect established this by considering the monic polynomial $p(t) = \prod_{i=0}^{n-1} (t - \lambda_i) = t^n - c_1 t^{n-1} - \cdots - c_n$. She constructed an explicit similarity transformation:

where $C(p)$ is the Frobenius companion matrix and $S$ is a lower-triangular matrix whose entries are derived from partial products of the nonpositive eigenvalues $\lambda_1, \dots, \lambda_{n-1}$. While constructive, the resulting matrix $A$ was generally asymmetric. Direct algebraic companion constructions were later simplified by Nizar Radwan (1996).

For more than two decades following Perfect's work, it remained open whether a Suleĭmanova spectrum could always be realized by a symmetric nonnegative matrix ($A = A^T \ge 0$).

In 1974, Miroslav Fiedler (1974) resolved this question affirmatively in Linear Algebra and its Applications, proving a cornerstone theorem of modern combinatorial matrix theory:

Fiedler's inductive proof constructs a sequence of symmetric matrices $A_1, A_2, \dots, A_n = A$ where at step $k$:

By selecting the coupling vector $\mathbf{v}_k \ge 0$ and the diagonal scalar $\alpha_k \ge 0$, the eigenvalues of $A_k$ interlace precisely with those of $A_{k-1}$, adjoining $\lambda_k \le 0$ without shifting the previous spectrum. Fiedler's theorem established that there is no symmetry gap for Suleĭmanova spectra:

In 1983, George W. Soules (1983) introduced a geometric paradigm shift: rather than constructing a bespoke matrix for each candidate spectrum, one can construct a single fixed orthogonal matrix $R \in \mathrm{O}(n)$ that diagonalizes all Suleĭmanova matrices simultaneously.

In 1998, Ludwig Elsner, Reinhard Nabben, and Michael Neumann (1998) characterized all Soules bases via binary tree sign structures:

Because Suleĭmanova's condition is an exact criterion for single-positive spectra, researchers have extended its mechanisms to broader configurations:

The Spectra Realizer tool on this research hub allows live synthesis of both asymmetric companion realizations and symmetric Fiedler/Soules matrices for any valid Suleĭmanova spectrum:

The publications below are sourced directly from the global NIEP Comprehensive Bibliography. Each record provides academic metadata, research notes, links to primary literature, and one-click BibTeX exports: