Spectracone in Nonnegative Matrix Theory

• Spectracone is a polyhedral cone defined by S such that S diag(x) S⁻¹ is nonnegative, representing realizable spectra for diagonalizable nonnegative matrices.
• It transforms the D-RNIEP into a system of linear inequalities, enabling a geometric and polyhedral approach to eigenvalue realization.
• The concept connects Perron similarities, row cones, Hadamard conic matrices, and character tables, offering unified insights into spectral realizability.

Searching arXiv for the cited papers on spectracones and related work. A spectracone is a polyhedral cone attached to an invertible matrix $S$, defined by the requirement that diagonal data become entrywise nonnegative after conjugation by $S$. In the real formulation used for the diagonalizable real nonnegative inverse eigenvalue problem (D-RNIEP), the spectracone is

$$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$

so that each $x\in\mathcal{C}(S)$ is a realizable eigenvalue list for a nonnegative matrix diagonalizable via the fixed similarity $S$. The notion was developed in the study of Perron similarities and Perron spectratopes, then refined through its relationship with row cones and row Hadamard conic matrices, and later extended to complex Perron similarities and ideal Perron similarities such as character tables of finite groups (Johnson et al., 2015, Johnson et al., 2016, Gershnik et al., 4 Aug 2025).

1. Definition and ambient framework

Let $S\in GL_n(\mathbb{R})$. A Perron similarity is an invertible matrix for which there exists a real diagonal nonscalar matrix $D$ such that $SDS^{-1}$ is entrywise nonnegative. In the 2015 formulation, this is equivalently characterized by the existence of an index $i\in\{1,\dots,n\}$ such that $Se_i\ge 0$ and $S$0. In the 2025 complex formulation, $S$1 is called a Perron similarity if there is a diagonal matrix $S$2 such that $S$3 is irreducible and nonnegative; equivalently, there is a unique $S$4 with

$S$5

where $S$6 satisfy $S$7 and $S$8 (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

Given such an $S$9, the spectracone collects all diagonal vectors that remain feasible under this similarity: $$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$0 in the real setting, and

$$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$1

in the complex setting, where $$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$2 is the diagonal matrix with diagonal entries $$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$3. The point $$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$4 may be regarded as a spectrum, because $$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$5 has eigenvalues $$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$6. Since $$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$7, the ray $$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$8 is always contained in the cone (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

The associated spectratope is a normalized slice of the cone, but the normalization depends on the framework. In the 2015 real setting,

$$\mathcal{C}(S)=\{x\in\mathbb{R}^n: S\operatorname{diag}(x)S^{-1}\ge 0\},$$9

whereas in the 2025 formulation,

$x\in\mathcal{C}(S)$0

Both constructions isolate normalized or stochastic realizations and convert the unbounded cone into a bounded polyhedral object (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

2. Relation to the nonnegative inverse eigenvalue problem

The spectracone is designed for the RNIEP and, more specifically, the D-RNIEP. If $x\in\mathcal{C}(S)$1, then

$x\in\mathcal{C}(S)$2

is diagonalizable and has spectrum $x\in\mathcal{C}(S)$3. Thus $x\in\mathcal{C}(S)$4 is exactly the set of eigenvalue lists realizable by nonnegative matrices that are diagonalizable via the fixed similarity $x\in\mathcal{C}(S)$5. The 2016 study makes this geometric role explicit: taken together, the spectracones of all Perron similarities constitute the solution to the diagonalizable RNIEP. Equivalently, the set of diagonalizable nonnegative spectra is the union of $x\in\mathcal{C}(S)$6 over all Perron similarities $x\in\mathcal{C}(S)$7 (Johnson et al., 2016).

The same point of view appears in normalized form in the earlier spectratope framework. Every normalized real spectrum of a diagonalizable nonnegative matrix lies in some Perron spectratope $x\in\mathcal{C}(S)$8. This rephrases the realization problem as a covering problem by polytopes subordinate to similarities, rather than as a direct search over nonnegative matrices (Johnson et al., 2015).

The symmetric nonnegative inverse eigenvalue problem is a special case of this framework. If $x\in\mathcal{C}(S)$9 is symmetric nonnegative, then it admits an orthogonal diagonalization

$S$0

so the relevant Perron similarity is orthogonal. In that case the spectracone becomes

$S$1

and it parameterizes spectra realizable by symmetric nonnegative matrices with the prescribed orthogonal eigenbasis (Johnson et al., 2016).

3. Row cones, row Hadamard conic matrices, and equality phenomena

A second cone naturally attached to $S$2 is the row cone

$S$3

where $S$4 denotes the $S$5-th row. Membership has a simple dual test: $S$6 entrywise. The comparison of $S$7 and $S$8 is a central structural problem in the theory of spectracones (Johnson et al., 2016).

The key algebraic notion controlling the inclusion $S$9 is the row Hadamard conic property. A matrix $S\in GL_n(\mathbb{R})$0 with rows $S\in GL_n(\mathbb{R})$1 is row Hadamard conic (RHC) if

$S\in GL_n(\mathbb{R})$2

where $S\in GL_n(\mathbb{R})$3 denotes the Hadamard product. For a Perron similarity $S\in GL_n(\mathbb{R})$4, one has the characterization

$S\in GL_n(\mathbb{R})$5

Equivalently, the row vectors themselves become spectral vectors exactly when the required Hadamard-product closure holds in the row cone (Johnson et al., 2016).

In the opposite direction, if some row of $S\in GL_n(\mathbb{R})$6 is the all-ones vector $S\in GL_n(\mathbb{R})$7, then

$S\in GL_n(\mathbb{R})$8

Combining both inclusions yields a practical equality criterion: if $S\in GL_n(\mathbb{R})$9 is a Perron similarity, some row equals $D$0, and $D$1 is RHC, then

$D$2

The 2016 paper also gives a more general characterization in terms of extremal vectors of $D$3: equality holds iff $D$4 is RHC and every extremal vector $D$5 of $D$6 satisfies $D$7 (Johnson et al., 2016).

Later work packages this coincidence into the notion of an ideal Perron similarity. In the 2025 terminology, $D$8 is ideal iff

$D$9

and this is equivalent to the conjunction of $SDS^{-1}$0 and the RHC property. In normalized settings, the same condition implies that the spectratope is the convex hull of the rows (Gershnik et al., 4 Aug 2025).

4. Polyhedral structure and explicit examples

The spectracone is polyhedral because the map $SDS^{-1}$1 is linear in the diagonal entries of $SDS^{-1}$2. If $SDS^{-1}$3 and $SDS^{-1}$4, then

$SDS^{-1}$5

so $SDS^{-1}$6 is a finite system of linear inequalities in $SDS^{-1}$7. In the real framework this yields a polyhedral cone in $SDS^{-1}$8; in the complex framework the same mechanism gives a polyhedral cone in $SDS^{-1}$9, with most applications concentrating on real $i\in\{1,\dots,n\}$0. The 2025 work additionally recalls that the spectracone is closed under Hadamard product (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

The geometry of $i\in\{1,\dots,n\}$1 can differ sharply from that of $i\in\{1,\dots,n\}$2. The 2016 paper gives explicit examples for all set-theoretic possibilities. For

$i\in\{1,\dots,n\}$3

entrywise nonnegativity is equivalent to $i\in\{1,\dots,n\}$4 and $i\in\{1,\dots,n\}$5, and one obtains

$i\in\{1,\dots,n\}$6

A second $i\in\{1,\dots,n\}$7 example gives the reverse proper inclusion. For

$i\in\{1,\dots,n\}$8

again with $i\in\{1,\dots,n\}$9 iff $Se_i\ge 0$0 and $Se_i\ge 0$1, one has

$Se_i\ge 0$2

hence

$Se_i\ge 0$3

A $Se_i\ge 0$4 example,

$Se_i\ge 0$5

produces spectracone generators

$Se_i\ge 0$6

for which neither $Se_i\ge 0$7 nor $Se_i\ge 0$8 holds, although the intersection is nontrivial. A common misconception is therefore that the spectracone and row cone typically coincide; the explicit examples show that proper containment in either direction, trivial intersection, and noncomparable intersection all occur (Johnson et al., 2016).

5. Structured families: Walsh matrices, Hadamard matrices, and character tables

The first major structured family arises from Hadamard matrices. For the Walsh matrix $Se_i\ge 0$9 of order $S$00, the 2015 paper proves that the Perron spectracone is exactly the conical hull of the rows: $S$01 Equivalently, if $S$02 is ordered $S$03, then

$S$04

Intersecting with the normalization $S$05 gives

$S$06

so the spectratope is a simplex. Explicit volume formulas are given: $S$07 This concrete geometry is strong enough to yield constructive realizations of normalized Suleĭmanova spectra by symmetric, doubly stochastic matrices whenever the dimension is a Hadamard order (Johnson et al., 2015).

At the same time, the Walsh behavior is not universal among Hadamard matrices. The 2015 paper notes that the “convex hull of rows” description does not hold for all Hadamard matrices, and for some Hadamard matrices of order $S$08, only the first row lies in the spectratope. This marks an important limitation: large determinant or Hadamard structure alone does not force ideality (Johnson et al., 2015).

A later and broader structured family is provided by character tables of finite groups. If $S$09 is the character table of a finite group $S$10, then $S$11 is a Perron similarity and, moreover, an ideal Perron similarity: $S$12 The proof uses the fact that the first row is $S$13 and that the Hadamard product of two character rows is the character of a tensor product representation, hence a nonnegative combination of irreducible characters. The same paper gives a group-theoretic inequality description of the spectracone: $S$14 When $S$15 is real, the projected spectratope of a diagonal rescaling is an $S$16-simplex, and its volume is

$S$17

These results unify earlier ideal examples: the discrete Fourier transform matrix is the character table of $S$18, Walsh matrices are character tables of $S$19, and Kronecker products of such matrices are character tables of direct products of groups (Gershnik et al., 4 Aug 2025).

6. Significance, variants, and open directions

The spectracone transforms a spectral existence problem into a problem in polyhedral geometry. For fixed $S$20, it replaces the search for nonnegative matrices with a finite system of linear inequalities in the coordinates of a diagonal vector. This yields a parameter space of diagonalizable nonnegative matrices,

$S$21

whose spectra are exactly the points of the cone. In favorable cases, especially when $S$22 is ideal, the geometry collapses to the row cone and the realizable spectra are generated directly by row data (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

The framework also interacts naturally with structural subclasses of the inverse eigenvalue problem. Orthogonal Perron similarities lead to the symmetric setting. In the character-table setting, if $S$23, then $S$24 is normal for all $S$25, while if $S$26 is real, then $S$27 is symmetric. Accordingly, the group-theoretic inequalities for $S$28 provide sufficient conditions for realizability in the normal or symmetric NIEP, depending on the table (Gershnik et al., 4 Aug 2025).

Several open directions are explicit in the literature. The 2015 work raises geometric extremality questions, including whether Walsh spectratopes have maximal volume among spectratopes subject to certain norm constraints. The 2025 work formulates a representation-theoretic conjecture: if $S$29 is a normalized ideal Perron similarity and is totally extremal, meaning all its entries are extremal points of the Karpelevič region, then $S$30 is the character table of a finite Abelian group. It also suggests extending the analysis to other structured matrices or representations, including induced representations and Hecke algebras (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

In this sense, the spectracone is not merely a convenient encoding of realizable spectra. It is a unifying object linking Perron–Frobenius theory, convex polyhedral geometry, Hadamard-product closure, association schemes, character theory, and multiple variants of the nonnegative inverse eigenvalue problem.