Ideal Perron Similarity

• Ideal Perron similarity is defined by the equality of a matrix’s spectracone and its row cone, simplifying spectrum realization in the nonnegative inverse eigenvalue problem.
• It leverages Hadamard closure and group-theoretic formulations—using matrices like Walsh, Fourier, and Vandermonde—to ensure that diagonalizable spectral lists are exactly generated by the matrix rows.
• The framework provides a polyhedral method for testing spectral feasibility, including totally extremal cases where rows form a Hadamard subgroup and represent extremal points in the L∞-unit ball.

An ideal Perron similarity is a Perron similarity $S$ for which the spectracone

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$

coincides with the conical hull of the rows of $S$, denoted $C_r(S)$. In that case, the diagonal spectra realized through the similarity $S$ are exactly the conical combinations of those rows; in the stochastic normalization, the corresponding spectratope is exactly the convex hull of the rows. The concept sits inside the diagonalizable nonnegative inverse eigenvalue problem, where one studies which spectral lists arise from entrywise nonnegative matrices via diagonal similarity models. Its geometric ingredients were developed in work on Perron spectratopes and row cones, and the terminology and complex-theoretic formulation were made explicit in later treatments (Johnson et al., 2015, Johnson et al., 2016, Johnson et al., 2024).

1. Definition within the nonnegative inverse eigenvalue problem

The nonnegative inverse eigenvalue problem asks which multisets occur as the spectrum of an entrywise nonnegative matrix. In the diagonalizable setting, one fixes an invertible matrix $S$ and considers matrices of the form

$A=S\Delta(x)S^{-1},$

with $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$. A Perron similarity is an invertible matrix for which at least one such diagonal choice yields an entrywise nonnegative realization; in the complex theory, $S\in GL_n(\mathbb C)$ is a Perron similarity if there exist an irreducible nonnegative matrix $A$ and a diagonal matrix $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$0 such that $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$1 (Johnson et al., 2024).

For a fixed $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$2, the central objects are the spectracone $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$3 and the spectratope $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$4. In the complex formulation,

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$5

Both are polyhedral: $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$6 is a polyhedral cone and $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$7 is a polytope. The row cone is

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$8

where $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$9 are the rows of $S$0, and the row polytope is $S$1 (Johnson et al., 2024).

An ideal Perron similarity is precisely a Perron similarity satisfying

$S$2

Equivalently, $S$3 is ideal if and only if $S$4 and every row $S$5 belongs to $S$6. In the stochastic slice, ideality is equivalent to

$S$7

This reformulates realizability by $S$8 in purely row-geometric terms (Johnson et al., 2024).

The importance of this notion is structural rather than merely terminological. For each Perron similarity, the associated cone and polytope provide a finite-dimensional polyhedral model of realizable spectra. The determination of the extremals of these convex sets for each Perron similarity would solve the diagonalizable nonnegative inverse eigenvalue problem (Johnson et al., 2024). This places ideal Perron similarities at a particularly tractable locus of the theory, because their extremal data are encoded directly by the rows.

2. Row cones, Hadamard structure, and exact criteria for equality

The real row-cone analysis gives the most explicit characterization of when $S$9 and $C_r(S)$0 coincide. For $C_r(S)$1, membership in the row cone has the simple test

$C_r(S)$2

This converts geometric questions about $C_r(S)$3 into linear inequalities involving $C_r(S)$4 (Johnson et al., 2016).

A key device is the Hadamard product. If $C_r(S)$5 denotes the $C_r(S)$6-th row and $C_r(S)$7, then

$C_r(S)$8

is equivalent to

$C_r(S)$9

This leads to the definition of a row Hadamard conic matrix: $S$0 is row Hadamard conic (RHC) if $S$1 for all $S$2. For Perron similarities, this is exactly the condition for the inclusion

$S$3

Thus the first half of ideality is equivalent to Hadamard closure of the rows inside the row cone (Johnson et al., 2016).

The converse inclusion is subtler. A sufficient condition is the presence of an all-ones row: if some row of $S$4 equals $S$5, then

$S$6

Under that hypothesis, ideality becomes equivalent to the RHC property. More generally, without assuming an all-ones row, equality holds if and only if two conditions are satisfied: $S$7 is RHC, and every extreme ray $S$8 of $S$9 obeys

$S$0

This gives a sharp criterion: one containment comes from Hadamard closure, the other from a finite extremal check because $S$1 is polyhedral (Johnson et al., 2016).

The same work exhibits all possible set-theoretic relations between $S$2 and $S$3: proper containment $S$4, proper containment $S$5, intersection only at the origin, and noncomparable cones with nontrivial intersection. A common misconception is therefore that row-generated behavior is automatic once $S$6 is a Perron similarity. It is not: ideality is a special equality, not a generic feature (Johnson et al., 2016).

3. Polyhedral structure, normalizations, and practical verification

For a fixed invertible $S$7, the inequalities defining $S$8 are explicit. In the complex theory, if $S$9 is the $A=S\Delta(x)S^{-1},$0-th row of $A=S\Delta(x)S^{-1},$1 and $A=S\Delta(x)S^{-1},$2 the $A=S\Delta(x)S^{-1},$3-th column of $A=S\Delta(x)S^{-1},$4, then

$A=S\Delta(x)S^{-1},$5

so nonnegativity of $A=S\Delta(x)S^{-1},$6 is equivalent to finitely many real-linear conditions. Consequently,

$A=S\Delta(x)S^{-1},$7

and $A=S\Delta(x)S^{-1},$8 is obtained by adding the affine constraints $A=S\Delta(x)S^{-1},$9 (Johnson et al., 2024).

This polyhedrality makes ideality algorithmically checkable. In the real setting, one may test whether $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$0 is a Perron similarity by solving the feasibility problem $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$1 and checking for a nonscalar solution. One may then test $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$2 through the RHC condition, namely by verifying

$\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$3

for all pairs $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$4. To test $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$5, one computes the extreme rays of $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$6 and checks $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$7 on each of them. If $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$8 has an all-ones row, the second step simplifies: ideality is then equivalent to the RHC condition alone (Johnson et al., 2016).

Several normalizations preserve or simplify the geometry. Left multiplication by a permutation matrix or by a positive diagonal matrix leaves both $\Delta(x)=\operatorname{diag}(x_1,\dots,x_n)$9 and $S\in GL_n(\mathbb C)$0 unchanged. Right multiplication by a permutation matrix permutes the coordinates of both cones; right multiplication by an invertible diagonal matrix leaves the spectracone unchanged while rescaling the row cone. In particular, if some row of $S\in GL_n(\mathbb C)$1 is totally nonzero, column scaling can convert it into an all-ones row without changing $S\in GL_n(\mathbb C)$2, thereby reducing the ideality check to the all-ones-row case (Johnson et al., 2016).

In the character-table setting, a further “one-column normalization” is standard: if $S\in GL_n(\mathbb C)$3, then $S\in GL_n(\mathbb C)$4 satisfies $S\in GL_n(\mathbb C)$5. This normalization is especially convenient for the stochastic slice, where $S\in GL_n(\mathbb C)$6 is cut by a coordinate hyperplane such as $S\in GL_n(\mathbb C)$7 (Gershnik et al., 4 Aug 2025).

4. Canonical families and constructive sources of ideality

The earliest systematic examples arose from Perron spectratopes of Hadamard matrices. For the canonical Walsh matrices $S\in GL_n(\mathbb C)$8, one has

$S\in GL_n(\mathbb C)$9

Thus Walsh matrices display the full row-generated behavior later formalized as ideality. These spectratopes also support constructive realizations of normalized Suleĭmanova spectra by symmetric doubly stochastic matrices (Johnson et al., 2015).

The discrete Fourier transform matrix $A$0 plays the analogous role in the complex theory. It is an ideal and extremal Perron similarity, $A$1, and its spectracone admits an explicit half-space description. Moreover, $A$2 diagonalizes circulants, and a spectrum $A$3 is realizable by an $A$4 circulant if and only if $A$5 (Johnson et al., 2024).

Vandermonde Perron similarities associated with Type I Karpelevich arcs furnish another major source. If the reduced Ito polynomial has distinct roots, the corresponding Vandermonde matrix $A$6 diagonalizes a nonnegative companion matrix realizing the arc, and $A$7 is ideal and extremal except for finitely many parameter values under the stated distinct-root hypotheses. These families generate large regions of realizable spectra and, together with $A$8 and a block-DFT construction, produce a nearly complete geometrical representation of spectra of $A$9 stochastic matrices (Johnson et al., 2024).

Kronecker products preserve ideality. If $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$00 and $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$01 are ideal Perron similarities, then $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$02 is again ideal. This extends classical Walsh constructions and produces mixed Fourier–Walsh families such as

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$03

which are ideal and extremal Perron similarities (Dockter et al., 2021, Johnson et al., 2024).

At the same time, ideality is not synonymous with belonging to a familiar matrix class. A notable counterexample is supplied by the normalized Hadamard matrix of order $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$04, for which only the all-ones row is realizable; hence $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$05. Likewise, nontriviality of $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$06 does not by itself imply that $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$07 is a Perron similarity (Johnson et al., 2024).

The symmetric nonnegative inverse eigenvalue problem fits into the same framework as the orthogonal special case. When $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$08 is orthogonal, $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$09 is symmetric, and the same cone criteria specialize verbatim to that setting (Johnson et al., 2016).

5. Character tables and the group-theoretic model

A decisive enlargement of the class of ideal Perron similarities came from character theory. If $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$10 is a finite group and $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$11 is its character table, then $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$12 is an ideal Perron similarity. This unifies several previously separate families: Walsh–Hadamard matrices are character tables of $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$13, discrete Fourier matrices are character tables of cyclic groups, and Kronecker products of these arise from direct-product groups (Gershnik et al., 4 Aug 2025).

The group-theoretic structure enters explicitly through inversion and inequality reduction. If $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$14 are conjugacy-class representatives and $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$15 are the irreducible characters, then

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$16

where $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$17. For $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$18,

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$19

Although this formula yields $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$20 entrywise constraints, the paper proves that nonnegativity is equivalent to only $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$21 scalar inequalities,

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$22

with real and imaginary parts separated when the character table is not real. This reduces spectracone membership to finitely many group-theoretic linear inequalities (Gershnik et al., 4 Aug 2025).

The reason character tables are ideal is closely related to the row Hadamard condition. The Hadamard product of two rows corresponds to the pointwise product of characters, and tensor-product decomposition expresses that product as a nonnegative integral combination of irreducible characters. Consequently, each row belongs to the spectracone, while the first row is the all-ones row, yielding

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$23

For real character tables, the projected spectratope after one-column normalization is an $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$24-simplex, and its Euclidean volume is given by

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$25

This is one of the rare closed-form geometric formulas presently available for projected Perron spectratopes (Gershnik et al., 4 Aug 2025).

A misconception dispelled by this development is that only Abelian Fourier-type matrices can be ideal. In fact, the character table of any finite group is ideal. The Abelian case becomes distinguished only when one imposes an additional extremality condition on every entry (Gershnik et al., 4 Aug 2025, Artemis et al., 1 Jun 2026).

6. Totally extremal ideality, Abelian classification, and open directions

The recent classification of totally extremal ideal Perron similarities isolates the sharp boundary case. A normalized ideal Perron similarity is required to satisfy

$C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$26

and it is called totally extremal when $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$27 for all $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$28. In this regime, the rows lie on the $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$29-unit sphere and are extreme points of the ambient $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$30-ball (Artemis et al., 1 Jun 2026).

The main structural result is that if $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$31 is a totally extremal normalized ideal Perron similarity, then its rows form a subgroup of $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$32 under the Hadamard product. Conversely, the rows of a nonsingular matrix form such a group if and only if the matrix is the character table of a finite Abelian group. Hence the following are equivalent: $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$33 is a totally extremal ideal Perron similarity, the rows of $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$34 form a Hadamard subgroup, and $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$35 is the character table of a finite Abelian group. This settles a conjecture of Gershnik, Laffey, and Paparella in the affirmative (Artemis et al., 1 Jun 2026).

In the Abelian case, admissible spectra in the stochastic slice are exactly Fourier transforms of probability distributions on the underlying group, and the corresponding matrices are group-circulant convolution matrices. The spectratope is the convex hull of the unimodular rows, and the paper interprets this as an extension of the Romanovsky–Karpelevič boundary phenomenon from individual unimodular eigenvalues to a full spectratope inside the $C(S)=\{x: S\Delta(x)S^{-1}\ge 0\}$36-unit ball (Artemis et al., 1 Jun 2026).

Several questions remain open. One is the classification of normalized ideal Perron similarities without the total extremality hypothesis; the Abelian character-table theorem does not settle that broader problem (Artemis et al., 1 Jun 2026). Another, already visible in the row-cone approach, asks whether every nonnegative spectrum occurs as a row of some RHC Perron similarity (Johnson et al., 2016). More globally, the program announced in the modern complex theory remains unresolved: determining the finitely many extremal spectra in the polyhedral sets attached to each Perron similarity would solve the diagonalizable nonnegative inverse eigenvalue problem (Johnson et al., 2024).

In this sense, ideal Perron similarity is both a definition and a strategy. It identifies those similarities for which the realizable spectra are exactly row-generated, converts realizability into polyhedral geometry, and links the inverse eigenvalue problem to Hadamard closure, representation theory, Fourier analysis, and the extremal geometry of stochastic spectra.