Characteristic-Vector Condition Overview

• Characteristic-Vector Condition is a framework for defining cross-disciplinary criteria based on characteristic data, unifying concepts from coding theory, spectral analysis, and linear rank inequalities.
• In coding theory, it condenses the structural information of a generator matrix into a multiplicity vector, enabling efficient reconstruction and weight-distribution computations with complexities like O(kq^k) in various settings.
• In spectral and matrix analysis, the condition appears as vanishing characteristic functions and Collatz–Wielandt criteria, providing a robust toolset for eigenvector determination and understanding obstruction phenomena.

Searching arXiv for the provided topic and cited papers. arxiv_search(query="Characteristic vector condition arXiv characteristic vector", max_results=10) Searching arXiv for the phrase and nearby formulations. {"query":"\"Characteristic vector\" arXiv linear code (Bouyukliev et al., 2018)", "max_results": 5} The expression Characteristic-Vector Condition is not used in a single standardized sense across the arXiv literature. Taken together, the sources suggest a family of context-dependent criteria in which some form of characteristic data—a projective multiplicity vector, a characteristic-function identity, a spectral condition, or a characteristic class—controls validity, reconstruction, existence of eigenvectors, or obstruction phenomena. In some papers the phrase does not occur literally and the relevant notion is identified only as the “closest formal criterion”; in another nearby line of work the relevant term is instead the characteristic endpoints condition for piecewise monotone maps (Bouyukliev et al., 2018, Gerasimova et al., 2010, Pena et al., 2019, Tang et al., 2021).

1. Terminological status and common pattern

Several sources explicitly state that they do not formulate a theorem literally called “Characteristic-Vector Condition,” but do isolate an exact nearest equivalent. In coding theory, that nearest equivalent is the requirement that a vector record projective column multiplicities of a generator matrix relative to a chosen simplex-code indexing. In characteristic-dependent rank inequalities, the nearest equivalent is a characteristic-sensitive projection identity for complementary subspaces, or a binary-matrix rank-jump criterion. In spectral theory of Jacobi matrices, the nearest equivalent is vanishing of a characteristic function together with the existence of an explicit recurrence solution in the operator domain. In nonlinear Perron–Frobenius theory, the relevant object is a positive eigenvector, also called a characteristic vector, with an exact Collatz–Wielandt criterion for existence and boundedness. This suggests that the phrase is best understood as a cross-disciplinary label for conditions extracted from characteristic data rather than as a single established theorem schema (Štampach, 2017, Stampach et al., 2012, Lins, 2021).

A plausible unifying interpretation is that such conditions typically compress a larger structure into a smaller invariant or functional datum and then recover a global consequence from that datum. The consequence may be algorithmic, as in weight-distribution computation; geometric, as in characteristic cycles; spectral, as in eigenvalue detection; or obstruction-theoretic, as in flatness and determinant-one criteria.

2. Coding theory: projective multiplicity vectors of linear codes

For a linear $[n,k]$ code $C$ over $\mathbb{F}_q$, with generator matrix $G$ having no zero columns and a fixed generator matrix $G_k$ of the $k$-dimensional simplex code $S_{q,k}$, the characteristic vector is

$$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$

where $x_u$ is the number of columns of $G$ that are equal or proportional to the $C$0-th column of $C$1. Equivalently, the entries are multiplicities of projective column classes in $C$2, indexed by the columns of $C$3. The exact validity criterion extracted in the source is that a vector is a valid characteristic vector precisely when its entries are nonnegative integers summing to $C$4 and are interpreted as those projective multiplicities. The source adds that from such a characteristic vector one can restore the columns of $C$5 up to order and multiplication by nonzero field elements (Bouyukliev et al., 2018).

This condition is computationally significant because it replaces the full generator matrix by a projective multiplicity function. If $C$6 and $C$7 denotes the $C$8-support matrix obtained by replacing every nonzero entry by $C$9, then the weights of a maximal set of nonzero projective codeword representatives are the coordinates of

$\mathbb{F}_q$0

In the binary case, after extending the characteristic vector to length $\mathbb{F}_q$1, the Walsh–Hadamard transform gives

$\mathbb{F}_q$2

while in the prime-field case the recursive array $\mathbb{F}_q$3 yields

$\mathbb{F}_q$4

The reported complexity is $\mathbb{F}_q$5, with $\mathbb{F}_q$6 in the binary case and $\mathbb{F}_q$7 for fixed prime $\mathbb{F}_q$8 in the prime-field case. The source also emphasizes that the characteristic vector is not an invariant of the abstract code alone: it depends on the chosen generator matrix $\mathbb{F}_q$9 and the chosen simplex-code generator matrix $G$0, although it is unchanged by column permutations of $G$1 (Bouyukliev et al., 2018).

A caveat, stated as implicit rather than formalized there, is that if one insists on recovering a rank-$G$2 generator matrix from such a multiplicity vector, the support must span $G$3. In the coding-theoretic setting of the paper this is automatic because the construction starts from an actual $G$4 code.

3. Characteristic-dependent rank inequalities: complementary subspaces and binary matrices

In the literature on characteristic-dependent linear rank inequalities, the clearest vector-space condition is a projection identity for complementary subspaces. Let $G$5 be mutually complementary in $G$6, and let $G$7 be such that $G$8 is a direct sum for every $G$9. Then the source proves

$G_k$0

where $G_k$1 denotes the canonical projection. This one-dimension gap is the explicit characteristic-sensitive mechanism from which the paper derives conditional and then unconditional characteristic-dependent linear rank inequalities (Pena et al., 2019).

The same source turns these conditional identities into unconditional inequalities by forcing general subspaces into a complementary configuration and paying codimension penalties measured by terms such as $G_k$2, $G_k$3, and $G_k$4. It then applies the resulting inequalities to network coding, proving that for each finite or co-finite set of primes $G_k$5 there exists a sequence of networks linearly solvable over a field if and only if the field characteristic lies in $G_k$6, while the linear capacity in the wrong characteristic tends to $G_k$7 as $G_k$8 (Pena et al., 2019).

A second paper makes the characteristic dependence even more explicit through binary matrices. If $G_k$9 is a binary matrix whose rank satisfies

$k$0

then the support patterns $k$1 can be converted into subspace-containment conditions that produce a pair of characteristic-dependent linear rank inequalities, one valid exactly when $k$2, the other valid exactly when $k$3. For each $k$4, the paper produces $k$5 such inequalities over $k$6 variables (Peña et al., 2019).

Taken together, these works suggest that in this area a “characteristic-vector condition” is best understood as a support-pattern or projection-pattern criterion whose dimension behavior changes when a specific integer becomes zero in the field.

4. Jacobi matrices: characteristic functions and explicit eigenvectors

For complex Jacobi matrices, the relevant condition is spectral and analytic. In the doubly infinite setting, a characteristic function $k$7 is built from the auxiliary function $k$8 under the summability condition

$k$9

for some $S_{q,k}$0. The source proves that, after regularization at isolated diagonal values, the zero set of $S_{q,k}$1 coincides with the point spectrum outside $S_{q,k}$2. More precisely, if

$S_{q,k}$3

on the natural analytic domain, then an eigenvector is given by the explicitly defined sequence $S_{q,k}$4, and the Wronskian identity

$S_{q,k}$5

shows that vanishing of the characteristic function is exactly the condition under which the canonical right- and left-subordinate solutions become linearly dependent and hence form a global $S_{q,k}$6-eigenvector. The same paper proves that the order of a zero of $S_{q,k}$7 equals the algebraic multiplicity of the corresponding eigenvalue, and that

$S_{q,k}$8

span the generalized eigenspace (Štampach, 2017).

In the one-sided Jacobi setting, the analogous construction produces a canonical vector

$S_{q,k}$9

with boundary term

$$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$0

The exact criterion is that

$$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$1

if and only if the distinguished recurrence solution $$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$2 is an $$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$3-eigenvector. On $$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$4, this is equivalent to $$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$5. The same source also identifies the finite-truncation characteristic polynomials and gives sufficient conditions under which spectra of truncated Jacobi matrices approximate the full spectrum (Stampach et al., 2012).

In this spectral literature, the “condition” is therefore not a combinatorial vector of multiplicities but a vanishing criterion for a characteristic function together with an explicit vector-valued recurrence solution.

5. Characteristic data in matrix, Gaussian, and nonlinear eigenvector problems

In the theory of self-congruence of bilinear forms, the closest characteristic-data criterion is spectral. For a nonsingular matrix $$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$6 over a field of characteristic different from $$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$7, the paper proves

$$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$8

In the singular case one must additionally exclude odd singular Jordan blocks $$x(C,G)=(x_1,\dots,x_{\theta(q,k)})\in \mathbb{Z}^{\theta(q,k)}, \qquad \theta(q,k)=\frac{q^k-1}{q-1},$$9. The source identifies this as the cleanest characteristic-data formulation: the determinant-one property is governed entirely by the $x_u$0-primary Jordan structure of the cosquare $x_u$1 (Gerasimova et al., 2010).

In the stability theory of the Ghurye–Olkin characterization, the relevant condition is a characteristic-function factorization for vector linear forms. After normalization to

$x_u$2

independence of $x_u$3 and $x_u$4 is equivalent to

$x_u$5

or, on logarithms,

$x_u$6

Under approximate factorization, the paper proves that the sum of the vectors in each equivalence class determined by equal matrix factors is near-Gaussian in the characteristic-function domain, and that every vector projection is near-Gaussian in the distribution-function domain (Mahvari et al., 2024).

In nonlinear Perron–Frobenius theory, a positive eigenvector is itself called a characteristic vector. For an order-preserving homogeneous map $x_u$7, the exact criterion is

$x_u$8

for every nonempty proper $x_u$9. The paper further observes that if the entries of $G$0 are real analytic, then nonemptiness and boundedness of the positive eigenspace are equivalent to uniqueness of the eigenvector up to scaling (Lins, 2021).

These examples indicate that outside coding theory the phrase often refers, in effect, to conditions on characteristic data—Jordan blocks, characteristic functions, or Collatz–Wielandt numbers—rather than to a single canonical vector object.

6. Vector bundles, characteristic rank, cleanliness, and universal characteristic classes

In vector-bundle topology, one nearby and established notion is characteristic rank. For a real vector bundle $G$1 over a connected finite CW-complex $G$2, the characteristic rank is the largest integer $G$3 such that every class

$G$4

is a polynomial in the Stiefel–Whitney classes $G$5. The bundle-dependent theory proves general criteria such as: if $G$6 is the first degree with nonzero reduced cohomology and $G$7, then $G$8; if $G$9, then $C$00. It also computes many exact values, including the cases of Dold manifolds, Moore spaces, stunted projective spaces, $C$01, and $C$02 (Naolekar et al., 2012).

A related paper computes the characteristic rank of vector bundles over Stiefel manifolds $C$03. There the controlling mechanism is the sparse mod-$C$04 cohomology together with Wu formulas and Steenrod operations. For $C$05 and $C$06, the paper determines the upper characteristic rank exactly as $C$07 and $C$08 in the stated ranges; in the real case it identifies the exceptional dimensions $C$09 as the only ones in which the first nonzero cohomology generator can be realized by an appropriate Stiefel–Whitney class (Korbaš et al., 2012).

In logarithmic $C$10-module geometry, the relevant condition is cleanliness. For a vector bundle with flat connection on $C$11, where $C$12 is a simple normal crossings divisor, cleanliness says roughly that singularities at closed points are controlled by the refined irregularity data at the generic points of $C$13. Under this condition, the paper proves

$C$14

so the logarithmic characteristic cycle is determined by the irregularity numbers and refined irregularities along the boundary components (Xiao, 2011).

A different but equally close notion appears in the recent construction of a universal characteristic class for vector bundles with connection. For a finite rank projective $C$15-module $C$16 and an $C$17-Lie-Rinehart algebra $C$18, the paper defines

$C$19

as the extension class of the fundamental exact sequence

$C$20

The central criterion is

$C$21

The paper further states that $C$22 is independent of the choice of connection, that $C$23 is an obstruction to algebraic parallelizability, and that in the given examples $C$24 is stronger than the Chern class and the Euler class (Maakestad, 28 Mar 2025).

Taken together, these vector-bundle and geometric works suggest a broad pattern: “characteristic” conditions may govern either the generation of cohomology by characteristic classes, the determination of characteristic cycles by refined local data, or the vanishing of a universal extension-class obstruction.