Perron Similarity in Matrix Spectral Theory

• Perron Similarity is a matrix-theoretic concept that organizes realizable spectra of nonnegative matrices via diagonal similarity and Perron–Frobenius theory.
• It employs both real and complex formulations, using spectracones and spectratopes to describe feasible eigenvalue sets in polyhedral terms.
• The framework supports constructive realizations of the nonnegative inverse eigenvalue problem, with models based on Hadamard, Walsh, and DFT matrices.

Perron similarity is a matrix-theoretic notion that organizes realizable spectra of nonnegative matrices through diagonal similarity. In the real formulation, an invertible matrix $S\in GL_n(\mathbb R)$ is a Perron similarity if there exists a real nonscalar diagonal matrix $D$ such that $SDS^{-1}\ge 0$ entrywise. In the complex formulation developed for the nonnegative inverse eigenvalue problem (NIEP), $S\in GL_n(\mathbb C)$ is a Perron similarity if it diagonalizes an irreducible nonnegative matrix $A=SDS^{-1}$. This viewpoint turns spectral realization into the study of the transfer map $x\mapsto M_x(S):=SD_xS^{-1}$, together with the associated spectracones and spectratopes that encode feasible eigenvalue vectors in polyhedral form (Johnson et al., 2015, Johnson et al., 2024).

1. Definition, normalization, and scope

The real and complex theories share the same core mechanism: one prescribes an invertible similarity $S$, places the desired eigenvalues on a diagonal matrix $D_x=\operatorname{diag}(x)$, and asks for entrywise nonnegativity of $M_x(S)=SD_xS^{-1}$. In the real RNIEP-oriented formulation, the defining requirement is the existence of a real nonscalar diagonal $D$ with $D$0. In the complex theory, the requirement is sharpened to the existence of an irreducible nonnegative matrix diagonalized by $D$1, which aligns the notion directly with Perron–Frobenius irreducibility (Johnson et al., 2015, Johnson et al., 2024).

Several refinements have become standard. The row cone of $D$2, denoted $D$3, is the conical hull of the rows of $D$4. An ideal Perron similarity is one for which the spectracone coincides with the row cone, $D$5. In the character-table literature, one further normalizes an ideal Perron similarity by requiring

$D$6

A normalized ideal Perron similarity is called totally extremal if every entry has unit modulus, equivalently $D$7 for all $D$8 (Gershnik et al., 4 Aug 2025, Artemis et al., 1 Jun 2026).

This framework sits inside the NIEP and its diagonalizable variants. If $D$9, then $SDS^{-1}\ge 0$0 is nonnegative and has spectrum given by the components of $SDS^{-1}\ge 0$1; if $SDS^{-1}\ge 0$2 lies in an appropriate normalized section $SDS^{-1}\ge 0$3, then $SDS^{-1}\ge 0$4 is stochastic. The theory is therefore both spectral and constructive: it simultaneously characterizes feasible lists and produces realizing matrices (Johnson et al., 2024).

2. Characterizations and structural criteria

The real theory gives a geometric characterization in terms of conical hulls of rows and columns. Writing the rows of $SDS^{-1}\ge 0$5 as $SDS^{-1}\ge 0$6 and the rows of $SDS^{-1}\ge 0$7 as $SDS^{-1}\ge 0$8, Johnson–Paparella proved that $SDS^{-1}\ge 0$9 is a Perron similarity if and only if there exists an index $S\in GL_n(\mathbb C)$0 such that

$S\in GL_n(\mathbb C)$1

For orthogonal matrices this collapses to a particularly simple test: $S\in GL_n(\mathbb C)$2 is a Perron similarity if and only if $S\in GL_n(\mathbb C)$3 entrywise for some $S\in GL_n(\mathbb C)$4 (Johnson et al., 2015).

The complex irreducible theory gives a Perron–Frobenius characterization at the level of a distinguished coordinate. An invertible $S\in GL_n(\mathbb C)$5 is a Perron similarity if and only if there exists a unique $S\in GL_n(\mathbb C)$6 such that

$S\in GL_n(\mathbb C)$7

where $S\in GL_n(\mathbb C)$8 satisfy $S\in GL_n(\mathbb C)$9 and $A=SDS^{-1}$0 are positive right and left Perron eigenvectors. This isolates the Perron eigendirection directly in the similarity matrix (Johnson et al., 2024).

A second structural layer concerns the relation between row geometry and nonnegativity. For $A=SDS^{-1}$1, row-cone membership has the exact test

$A=SDS^{-1}$2

If $A=SDS^{-1}$3 denotes the $A=SDS^{-1}$4-th row of $A=SDS^{-1}$5, then $A=SDS^{-1}$6 is row Hadamard conic (RHC) when $A=SDS^{-1}$7 for all $A=SDS^{-1}$8, where $A=SDS^{-1}$9 is the Hadamard product. In the real theory, $x\mapsto M_x(S):=SD_xS^{-1}$0 holds exactly when $x\mapsto M_x(S):=SD_xS^{-1}$1 is RHC, and if some row of $x\mapsto M_x(S):=SD_xS^{-1}$2 is $x\mapsto M_x(S):=SD_xS^{-1}$3, then $x\mapsto M_x(S):=SD_xS^{-1}$4; under that normalization, equality $x\mapsto M_x(S):=SD_xS^{-1}$5 is equivalent to the RHC property (Johnson et al., 2016). In the later complex ideality criterion, this is recast as: $x\mapsto M_x(S):=SD_xS^{-1}$6 is ideal if and only if $x\mapsto M_x(S):=SD_xS^{-1}$7 and $x\mapsto M_x(S):=SD_xS^{-1}$8 is RHC (Gershnik et al., 4 Aug 2025).

3. Spectracones, spectratopes, and polyhedral geometry

For an invertible similarity $x\mapsto M_x(S):=SD_xS^{-1}$9, the fundamental feasibility set is the spectracone

$S$0

The normalization used for the associated spectratope depends on the setting. In the original real paper one uses

$S$1

whereas the complex stochastic theory uses

$S$2

Both constructions define a bounded polyhedral slice of the spectracone, but they encode different normalizations: fixed first coordinate in the real RNIEP treatment, and stochasticity in the complex treatment (Johnson et al., 2015, Johnson et al., 2024).

Polyhedrality is explicit. In the real setting, the entrywise inequalities are

$S$3

and vectorization gives a linear system $S$4, where

$S$5

with $S$6 the diagonal-selection matrix whose columns are $S$7. Thus $S$8 is a polyhedral cone, and $S$9 is a bounded polytope obtained by slicing that cone (Johnson et al., 2015). In the complex theory, $D_x=\operatorname{diag}(x)$0 and $D_x=\operatorname{diag}(x)$1 are likewise polyhedral when $D_x=\operatorname{diag}(x)$2 is viewed as a $D_x=\operatorname{diag}(x)$3-dimensional real space; moreover, $D_x=\operatorname{diag}(x)$4 and $D_x=\operatorname{diag}(x)$5 are closed under the Hadamard product, and the realizing family

$D_x=\operatorname{diag}(x)$6

is a convex cone closed under matrix multiplication (Johnson et al., 2024).

These polyhedra organize the diagonalizable NIEP. If $D_x=\operatorname{diag}(x)$7 is diagonalizable with real spectrum $D_x=\operatorname{diag}(x)$8 and $D_x=\operatorname{diag}(x)$9, then $M_x(S)=SD_xS^{-1}$0, and after normalizing by the Perron eigenvalue one obtains a point of $M_x(S)=SD_xS^{-1}$1. Consequently, the set of normalized real spectra of diagonalizable nonnegative matrices is covered by the union of Perron spectratopes over Perron similarities. In the complex stochastic theory, the extremals of $M_x(S)=SD_xS^{-1}$2 and $M_x(S)=SD_xS^{-1}$3 are finite in number for each fixed $M_x(S)=SD_xS^{-1}$4, and determining them for every Perron similarity would solve the diagonalizable NIEP, described there as a major portion of the entire problem (Johnson et al., 2015, Johnson et al., 2024).

4. Canonical families and exact models

Hadamard and Walsh matrices provide the classical real model. For the canonical Walsh matrix $M_x(S)=SD_xS^{-1}$5 of order $M_x(S)=SD_xS^{-1}$6, the Perron spectracone is exactly the conical hull of the rows,

$M_x(S)=SD_xS^{-1}$7

and the Perron spectratope is exactly the convex hull of the rows,

$M_x(S)=SD_xS^{-1}$8

The associated realizations admit an association-scheme description: if $M_x(S)=SD_xS^{-1}$9, then

$D$0

decomposes as a nonnegative combination of trisymmetric permutation matrices $D$1, and the family $D$2 forms a $D$3-class association scheme (Johnson et al., 2015).

The complex counterpart is the discrete Fourier transform matrix $D$4. Its spectracone and spectratope coincide with the conical and convex hulls of its rows, respectively, so $D$5 is ideal and extremal. Moreover, $D$6 is circulant, and feasibility has the exact linear test

$D$7

Accordingly, $D$8 is realizable by a nonnegative circulant matrix if and only if $D$9. The same ideality and extremality persist for multifold Kronecker products of DFT matrices and for mixed Kronecker products of DFT and Walsh matrices (Johnson et al., 2024).

Character tables extend ideality from abelian to arbitrary finite groups. If $D$00 is the character table of a finite group, then $D$01 is an ideal Perron similarity. Its spectracone is described by finitely many group-theoretic inequalities: $D$02 When $D$03 is real, the projected spectratope is a simplex with an explicit volume formula in terms of centralizer sizes and irreducible character degrees (Gershnik et al., 4 Aug 2025).

The most rigid case is now classified. Totally extremal normalized ideal Perron similarities are precisely the character tables of finite abelian groups. Equivalently, for $D$04, the rows form a group under Hadamard product if and only if $D$05 is the character table of a finite abelian group, and that row-group is canonically isomorphic to the underlying group (Artemis et al., 1 Jun 2026).

Kronecker products preserve much of this structure. If $D$06 and $D$07 are Perron similarities, then $D$08 is a Perron similarity; if $D$09 and $D$10 are ideal, then $D$11 is ideal. The induced cones satisfy

$D$12

and these inclusions are generally strict (Dockter et al., 2021).

5. Constructive realizations and the NIEP program

Perron similarities are not only classificatory objects; they provide explicit realizations. In the Hadamard setting, the 2015 real theory gives a constructive version of Fiedler’s theorem at Hadamard orders: if $D$13 is a normalized Hadamard matrix of order $D$14 and $D$15 is a normalized Suleĭmanova spectrum, then $D$16 is realizable by a symmetric, doubly stochastic matrix. The construction writes the target vector $D$17 as a convex combination of $D$18 and the vectors $D$19, shows $D$20, and then forms

$D$21

A further corollary states that after adjoining sufficiently many zeros, a normalized Suleĭmanova spectrum is realized by a symmetric, doubly stochastic matrix of larger order; if the enlarged order is a power of two, the realizing matrix may be chosen trisymmetric (Johnson et al., 2015).

The complex theory connects Perron similarities to boundary constructions in the Karpelevich region. For Type I Karpelevich arcs, the reduced Ito polynomial

$D$22

has a nonnegative irreducible companion realization $D$23. When the roots are distinct, the associated Vandermonde matrix $D$24 satisfies

$D$25

so $D$26 is a Perron similarity. These Type I realizers generate large portions of the stochastic spectral region. For $D$27 stochastic matrices, spectratopes coming from Type I realizers, from $D$28, and from the mixed similarity

$D$29

with

$D$30

yield a nearly complete geometrical representation of the spectra of $D$31 stochastic matrices (Johnson et al., 2024).

This constructive role explains the importance of ideal families. When $D$32 is ideal, the extreme rays and extreme points of the feasible spectral region are the rows of $D$33 and their normalized convex combinations, so realizability reduces to explicit polyhedral combinatorics rather than an unrestricted search over diagonal conjugations (Gershnik et al., 4 Aug 2025, Johnson et al., 2024).

6. Limitations, terminology, and current directions

Several caveats are built into the theory. First, the Perron and ideal properties are independent: a matrix can be ideal but not a Perron similarity, and conversely. Second, the union of Perron spectratopes captures normalized spectra of diagonalizable nonnegative matrices, but not necessarily all spectra of nonnegative matrices; non-diagonalizable cases are not directly represented as $D$34. Third, in the real polyhedral formulation, numerical testing through the linear feasibility system $D$35 can be sensitive to the conditioning of $D$36 and to the combinatorial structure of the selection matrix $D$37. The 2015 paper also notes that covering trace-nonnegative polytopes may require uncountably many spectratopes in dimension $D$38, so no finite catalog of Perron similarities suffices even at low dimension (Artemis et al., 1 Jun 2026, Johnson et al., 2015).

A recurrent misconception concerns “nontriviality” of the spectracone. A later paper gave a counterexample to the stronger claim that

$D$39

is equivalent to $D$40 being a Perron similarity. The counterexample exhibits a matrix $D$41 for which $D$42 properly contains $D$43 but no column of $D$44 and corresponding row of $D$45 share a global nonnegative or nonpositive sign, so $D$46 is not a Perron similarity in the sign-criterion sense used there (Dockter et al., 2021).

There is also a genuine terminological fork. In a separate line of work on matrices with negative entries, “Perron similarity” refers to similarity to an eventually positive matrix. In that setting, if $D$47 has a positive right eigenvector $D$48, a real left eigenvector $D$49, and a positive simple dominant eigenvalue $D$50, then

$D$51

is an explicit conjugate, and $D$52 has a positive, simple, dominant eigenvalue if and only if $D$53 is eventually positive. For row-stochastic matrices this yields the entrywise criterion

$D$54

for some $D$55 and all $D$56 (Labbé et al., 2015). This usage is related by similarity and Perron–Frobenius behavior, but it is distinct from the spectracone-based NIEP literature.

Current directions follow the existing classification results. One strand asks for a characterization of polyhedral cones or polytopes closed under Hadamard product that arise as $D$57 or $D$58. Another studies normalized ideal Perron similarities beyond the totally extremal case, where the abelian character-table classification no longer applies directly. A further boundary question in the stochastic spectral region asks whether every boundary point of $D$59 is extremal for $D$60 (Johnson et al., 2024, Artemis et al., 1 Jun 2026).