Cyclotomic Matrix: Properties and Applications

• Cyclotomic matrices are highly structured matrices defined by entries and spectra derived from the partitioning of roots of unity, with applications in algebraic number theory and coding theory.
• They are constructed using finite field methods, character-based formulations, and classical integer approaches, with properties linked to Gauss sums, circulant forms, and explicit determinant formulas.
• These matrices enable practical insights for analyzing difference sets, cryptographic algorithms, and combinatorial identities, bridging computational methods and theoretical arithmetic.

A cyclotomic matrix is a highly structured matrix whose entries, spectrum, and algebraic properties encode deep arithmetic and combinatorial relationships arising from cyclotomy—the partitioning of roots of unity, finite field elements, or integer residues into cosets induced by subgroups of their respective multiplicative groups. These matrices occur throughout algebraic number theory, finite geometry, coding theory, and the paper of character sums, often serving as linear-algebraic avatars for cyclotomic numbers, Jacobi and Gauss sums, and Schur ring structures.

1. Foundational Definitions and Constructions

The term “cyclotomic matrix” admits several formulations depending on context: over finite fields, over rings of algebraic integers, and as concrete realizations of cyclotomic numbers.

• Finite Field Construction: For $F_q$ a finite field of characteristic $p>2$, let $\ell$ be a divisor of $q-1$ and $K = \{x^\ell: x \in F_q^\times\}$ the subgroup of $\ell$th powers. The cyclotomic matrix $A$ of order $\ell$ is given entrywise by $A_{i,j} = |(1 + g^i K) \cap g^j K|$, capturing cyclotomic numbers $(i,j)$ as intersection sizes of cosets in $F_q^\times$ (Sun, 17 Nov 2025).
• Character-Based Matrices: Over $F_q$, fix a multiplicative character $\chi$. Cyclotomic matrices often take the form $$B_{q,k}(\chi^r) = [\chi^r(a_i + a_j) + \chi^r(a_i - a_j)]$$, with $a_i$ parametrizing $k$th power cosets. For $k=2$, $a_i$ runs over nonzero squares, leading to explicit quadratic-type cyclotomic matrices (Wu et al., 30 Jul 2024).
• Classical Integer and Algebraic Integers: A cyclotomic matrix $A$ over a ring $R \subset \mathbb{C}$ is Hermitian, with integer characteristic polynomial $\chi_A(x)$ and spectrum in $[-2,2]$. Kronecker's theorem ensures that $z^n \chi_A(z + z^{-1})$ factors into cyclotomic polynomials, and all eigenvalues are of the form $\mu + \mu^{-1}$ for roots of unity $\mu$ (Greaves, 2012, Greaves, 2011).
• Matrices of Cyclotomic Numbers: Given $n$ and a square-free $e$, matrices such as $A = ((a,b)_e)$ (with $(a,b)_e$ the cyclotomic number of order $e$) are studied in combinatorics and algebraic number theory (Ahmed et al., 2018, Sun, 17 Nov 2025).

2. Spectral, Determinant, and Structural Properties

Cyclotomic matrices possess intricate spectral properties governed by their underlying group or field structures. Determinant formulas frequently involve character sums, Gauss sums, or combinatorial invariants.

• Spectrum and Minimal Polynomial: Cyclotomic matrices over rings of algebraic integers are classified by their spectra in $[-2,2]$ and by integrality of the characteristic polynomial. The spectrum of structured block matrices associated to power difference sets is explicitly described via the parameters $k, \lambda$ and their combinatorial relationships (Greaves, 2012, Sun, 17 Nov 2025).
• Determinant Formulas via Character Sums: For matrices indexed by cosets of squares, determinants often factor into products of Jacobi or Gauss sums, sometimes collapsing to explicit expressions in terms of hypergeometric functions over finite fields. For $B_{q,2}(\chi^r)$ with $q \equiv 3 \pmod{4}$, $\det B_{q,2}(\chi^r)$ factors as $\prod_{k=0}^{(q-3)/2} J_q(\chi^r,\chi^{2k})$ and admits closed forms involving $G_q(\chi^r)$ (Wu et al., 30 Jul 2024).
• Singularity Criteria: Invariant and singularity conditions arise, e.g., $B_q((q-3)/2)$ becomes singular for $q = p^f$ with $f \ge 2$ due to binomial coefficient vanishing mod $p$ via Lucas's theorem, or specific Pell sequence congruence in prime fields (Wu et al., 3 Jan 2025).
• Explicit Circulant and Trinomial Cases: Variants involving trinomial coefficients (e.g., $S_q(i,j) = \chi(a_i^2 + a_i a_j + a_j^2)$) yield circulant matrices whose determinants connect directly to central trinomial numbers and singularity to explicit polynomial vanishing (Li et al., 2022).

3. Cyclotomic Matrices and Classical Arithmetic

Cyclotomic matrices serve as concrete arithmetic representations of number-theoretic phenomena.

• Connection to Jacobi and Gauss Sums: Throughout, matrices with entries $J_p(\chi^{ki},\chi^{kj})$ or $G_q(\chi^{r})$ encapsulate the structure of Jacobi sums, the algebraic integers generated by roots of unity, and the associated minimal polynomials $P_k(T)$ for cyclotomic units (Wu et al., 17 Jun 2025, Wu et al., 23 Apr 2024).
• Schur Ring and Group Ring Structures: Embedding cyclotomic matrices into group rings or Schur rings leads to structural identities and product rules such as $$A_u A_v = k[\delta_{u,v+q'} I_\ell - E_{u+q',v}] + \sum_w (u-v, w-v) A_w$$ which underlie combinatorial relations among cyclotomic numbers (Sun, 17 Nov 2025).
• Power Difference Sets: Cyclotomic matrices formalize difference set conditions: a subset $K$ of order $\ell$ powers forms a $(q,k,\lambda)$-difference set if entries $(i,0) = \lambda$ for all $i$, directly verifiable by cyclotomic matrix formulation (Sun, 17 Nov 2025).

4. Circulant and Vandermonde Cyclotomic Matrices

Circulant and Vandermonde cyclotomic matrices play a key role in algebraic number theory and cryptography.

• Circulant Representations of Cyclotomic Fields: Every subfield of the $p$th cyclotomic field $\mathbb{Q}(\zeta_p)$ is represented by a unique $p\times p$ $0$-$1$ circulant matrix built from the subgroup structure of $(\mathbb{Z}/p\mathbb{Z})^\times$. Matrix-algebra representations directly realize the field extension structure (Reddy et al., 2011).
• Vandermonde Matrix of Roots of Unity: The Vandermonde matrix $V_n$ formed from primitive $n$th roots $\zeta_i$ is central in structure theory. The condition number $\operatorname{Cond}(V_n)$ depends only on the radical of $n$ and, for $n=2^k p^\ell$, is explicitly $$\operatorname{Cond}(V_n) = 2^{k-1} p^{\ell-1} (p-1) = \frac{2^k p^\ell}{\varphi(2^k p^\ell)}$$ (Scala et al., 2020).
• Cryptographic Significance: For $n$ of the form $2^k p^\ell$, upper bounds on the condition number of $V_n$ guarantee the polynomial equivalence of RLWE and Poly-LWE problems, imposing only polynomial distortion on error distributions (Scala et al., 2020).

5. Cyclotomic Matrices over Rings of Quadratic and Imaginary Integers

Classifications of cyclotomic matrices over rings such as $\mathbb{Z}[\sqrt{d}]$, $\mathbb{Z}[i]$, or $\mathbb{Z}[\omega]$ reveal discrete families and sporadic examples.

• Characterization by Spectrum and Polynomials: Hermitian matrices with spectrum in $[-2,2]$ and integer characteristic polynomials are fully classified into infinite families—toral, cylinder, and charged cylinder graphs—and sporadic graphs, with explicit adjacency and weight rules depending on ring structure (Greaves, 2012, Greaves, 2011).
• Proof Techniques: Cauchy interlacing, Perron–Frobenius theory, and Gram-matrix decompositions are systematically used to enumerate all indecomposable cases, eliminate forbidden subgraphs, and establish structural constraints on degrees and edge weights.

6. Inversion and Orthogonality of Cyclotomic Matrices

Matrices built from trigonometric expressions of cyclotomic nature possess explicit inversion formulas governed by group-theoretic arithmetic.

• Sine and Cosine Cyclotomic Matrices: For square-free $n$ and $R = \{j : 1 \le j \le n/2, \gcd(j,n)=1\}$, the matrices $S_{j,k} = 2\sin(2\pi j k^*/n)$ and $C_{j,k} = 2\cos(2\pi j k^*/n)$ are invertible precisely when $n$ is square-free (or $n=4$ for sine), with inverses $\hat S_{j,k}$ and $\hat C_{j,k}$ given by explicit arithmetic functions involving residue classes mod $n$ (Girstmair, 2018). Gauss sum theory underpins the diagonalization and orthogonality of these matrices.

7. Analogies, Generalizations, and Contemporary Applications

Cyclotomic matrices unify diverse domains through analogies to classical theory and contemporary applications.

• Hypergeometric Function Analogues: Determinants and eigenvalue formulas often have finite-field hypergeometric function representations—Greene's finite-field analogues of $_{n+1}F_n$—facilitating connections to point-counts on algebraic varieties and supercharacter theory (Wu et al., 2021, Wu et al., 30 Jul 2024).
• Gamma Function Determinant Parallels: The determinant properties of Gauss sum matrices in finite fields mirror those of gamma-function matrices in complex analysis, supported by analogues of the Hasse–Davenport product formula (Wu et al., 23 Apr 2024).
• Singularity, Square Conditions, and Zeta Functions: Determinant square conditions resolved via cyclotomic matrices confirm conjectures about arithmetic invariants and regulate factorization of zeta functions of curves over finite fields (Wu et al., 2021, Wu et al., 17 Jun 2025).

Cyclotomic matrices thus form a nexus for arithmetic, combinatorial, and spectral phenomena, providing both explicit computational tools and conceptual bridges among foundational topics in modern algebra and number theory.