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The Fibonacci--Redheffer matrix and its properties

This paper defines a Redheffer-type matrix with Fibonacci entries, analyzes its determinant and spectral properties, explores generalizations with number-theoretic examples, and presents new asymptotic results alongside a novel expression related to the Riemann hypothesis.

Original authors: Aristides V. Doumas, Panayiotis J. Psarrakos

Published 2026-04-08

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Aristides V. Doumas, Panayiotis J. Psarrakos

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery involving a giant, complex grid of numbers. This grid is called a matrix. In the world of mathematics, these grids are like spreadsheets that hold secrets about how numbers interact.

This paper introduces a new, special kind of grid called the Fibonacci–Redheffer matrix. To understand it, let's break it down into a story using some everyday analogies.

1. The Original Grid: The "Divisor" Map

First, the authors talk about an old, famous grid invented in 1977 by a man named Redheffer. Imagine this grid as a social network map for the numbers 1 through $n$ .

The Rule: If number $A$ is a "divisor" of number $B$ (meaning $B$ can be divided by $A$ without a remainder), they get a "friendship" mark (a 1) in the grid.
The Special Case: Number 1 is friends with everyone, so the first column is all 1s.
The Mystery: For decades, mathematicians used this grid to study the Riemann Hypothesis, one of the biggest unsolved puzzles in math. The "total score" (determinant) of this grid tells us something deep about prime numbers.

2. The New Grid: The "Fibonacci" Twist

The authors of this paper decided to spice things up. They took the same social network map but changed the rules for the "friendship marks."

Instead of just putting a "1" for every friendship, they put in Fibonacci numbers.
What are Fibonacci numbers? They are a famous sequence where each number is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21... They grow very fast, like a rabbit population.
The New Rule: If number $A$ $A$ is a divisor of $B$ $B$ , the grid doesn't just say "1"; it says, "How big is the Fibonacci number for $A$ $A$ ?"
- If 2 divides 4, the grid puts in the 2nd Fibonacci number (which is 1).
- If 3 divides 6, the grid puts in the 3rd Fibonacci number (which is 2).
- If 5 divides 10, the grid puts in the 5th Fibonacci number (which is 5).

This creates a Fibonacci–Redheffer matrix. It looks like the old grid, but the numbers are much bigger and grow rapidly.

3. What Did They Discover?

The authors spent their time analyzing this new grid to see what happens when you crunch the numbers. Here are their main findings, translated into plain English:

A. The "Total Score" is Always Negative

In math, every grid has a "total score" called the determinant. For the old Redheffer grid, this score jumps around wildly (sometimes positive, sometimes negative, sometimes zero).

The Discovery: For the new Fibonacci grid, the total score is always negative (for grids larger than 2x2). It's like a scale that always tips to the left, no matter how many numbers you add.
Why it matters: This tells us the grid is "stable" in a specific way; it never collapses to zero (which would mean the grid is useless for solving equations).

B. The "Vibrations" (Eigenvalues)

Imagine the grid is a drum. If you hit it, it vibrates at specific frequencies. In math, these frequencies are called eigenvalues.

The Discovery:
1. Real and Simple: All the vibrations are "real" (not imaginary) and distinct (no two are exactly the same).
2. The "Ghost" Vibration: There is exactly one vibration that is negative (a "ghost" in the machine). All the others are positive.
3. The Fibonacci Connection: The positive vibrations are almost exactly the same as the Fibonacci numbers themselves!
  - The 2nd vibration is just a tiny bit bigger than the 2nd Fibonacci number.
  - The 3rd vibration is just a tiny bit bigger than the 3rd Fibonacci number.
  - It's as if the grid is trying to hum the Fibonacci song, but slightly out of tune.

C. The "Messengers" (Eigenvectors)

Every vibration has a "messenger" (an eigenvector) that carries the energy across the grid.

The Discovery: In the old grid, some messengers were lazy and had zeros in their pockets (zero entries). In this new Fibonacci grid, every single messenger is active. Every number in the messenger's list is non-zero. The energy flows everywhere; nothing is left out.

4. The Bigger Picture

The authors also looked at what happens if you make the grid infinitely large.

They found that the "total score" (determinant) grows incredibly fast, following a specific mathematical curve related to the Fibonacci numbers.
They showed that this idea isn't just about Fibonacci numbers. You can swap the Fibonacci numbers for any other sequence of numbers (like powers or logarithms) and get similar, fascinating results.

The Takeaway

Think of this paper as a musical experiment.

The Redheffer matrix is an old, slightly chaotic instrument.
The Fibonacci–Redheffer matrix is a new, upgraded instrument where the strings are tuned to the famous Fibonacci sequence.
The authors discovered that this new instrument is perfectly tuned: it never goes silent (non-zero determinant), it plays a clear, distinct note for every string (simple eigenvalues), and every note is closely related to the Fibonacci sequence itself.

This helps mathematicians understand how structured patterns (like the Fibonacci sequence) influence the hidden "music" of numbers, which might eventually help solve even bigger mysteries in the world of math.

1. Problem Statement

The paper investigates the spectral properties and determinant of a novel matrix structure called the Fibonacci–Redheffer matrix, denoted as $FR(n)$. This matrix is a modification of the classical Redheffer matrix ( $R_n$ ), which is defined by $r_{i,j} = 1$ if $i|j$ or $j=1$ , and $0$ otherwise. The Redheffer matrix is famous for its connection to the Riemann Hypothesis via the Mertens function ( $M(n) = \det(R_n)$ ).

The authors introduce $FR(n)$ by replacing the $1$s in the $i$ -th row (where $i|j$ ) with the $i$ -th Fibonacci number ( $F_i$ ), while retaining the $1$s in the first column. The core problem is to determine:

The exact value and asymptotic behavior of $\det(FR(n))$ .
The nature of the eigenvalues (real vs. complex, multiplicity) and eigenvectors of $FR(n)$.
How the rapid growth of the Fibonacci sequence influences the spectral distribution compared to the classical Redheffer matrix.
Generalizations of these properties to other sequences.

2. Methodology

The authors employ a combination of linear algebra techniques, number theory, and asymptotic analysis:

Matrix Decomposition: The matrix $FR(n)$ is decomposed into $FR(n) = C(n) + D(n)$, where $D(n)$ is an upper triangular matrix containing Fibonacci numbers on the diagonal (and multiples thereof), and $C(n)$ is a rank-one singular matrix.
Inverse and Determinant Calculation: The inverse of the triangular matrix $D(n)$ is explicitly derived using the Möbius function $\mu(k)$ . The determinant of $FR(n)$ is then computed using the matrix determinant lemma (or rank-one perturbation formula) applied to $D(n) + C(n)$ .
Spectral Analysis via Characteristic Functions: To analyze eigenvalues, the authors construct a rational function $Q_{FR(n)}(z)$ derived from the characteristic equation. They analyze the asymptotes and limits of this function to locate roots (eigenvalues) relative to the Fibonacci numbers.
Rank and Multiplicity Proofs: Lemmas are used to prove that the geometric multiplicity of any eigenvalue is 1 and that eigenvectors have no zero entries. This involves contradiction arguments assuming a zero entry in an eigenvector.
Asymptotic Analysis: Theorems regarding the Fibonacci factorial ( $n!_F$ ) and the convergence of series involving $\mu(k)/F_k$ are used to derive asymptotic formulas for the determinant and spectral radius as $n \to \infty$ .
Generalization: The framework is extended to a general class of Redheffer-type matrices $A_R(n)$ defined by an arbitrary sequence $\alpha = \{a_j\}$ .

3. Key Contributions and Results

A. Determinant Properties

Exact Formula: The determinant is derived as:
$\det(FR(n)) = n!_F \sum_{k=1}^n \frac{\mu(k)}{F_k}$
where $n!_F = \prod_{i=1}^n F_i$ is the Fibonacci factorial.
Sign: For all $n \geq 3$ , the determinant is strictly negative. This contrasts with the classical Redheffer matrix, whose determinant (the Mertens function) oscillates and has infinitely many zeros.
Asymptotics: As $n \to \infty$ , the determinant behaves as:
$\det(FR(n)) \sim C_0 \frac{\phi^{n(n+1)/2}}{5^{n/2}}$
where $\phi$ is the golden ratio and $C_0 \approx -0.792$ . The series $\sum \frac{\mu(k)}{F_k}$ converges to a negative constant $C \approx -0.6457$ .

B. Spectral Properties

Real and Simple Eigenvalues: All $n$ eigenvalues of $FR(n)$ are real and simple (algebraic multiplicity 1).
Non-zero Eigenvectors: Every eigenvector of $FR(n)$ has no zero entries. This is a significant departure from the classical Redheffer matrix, which has eigenvectors with zero entries and is not diagonalizable for $n \geq 5$ .
Eigenvalue Localization:
- There is exactly one negative eigenvalue, $\lambda_1$ , satisfying $-1 < \lambda_1 < 0$ .
- The remaining $n-1$ eigenvalues are positive and interlace with the Fibonacci numbers. Specifically, for $i = 2, \dots, n-1$ :
  $F_i < \lambda_i < F_{i+1}$
- For large $n$ , the largest eigenvalue $\lambda_n$ satisfies $\lambda_n > F_n$ .
Spectral Radius: The spectral radius grows asymptotically as $\rho(FR(n)) \sim \frac{\phi^n}{\sqrt{5}}$ , which is the growth rate of the Fibonacci sequence itself.

C. Generalizations

The paper generalizes the results to matrices $A_R(n)$ defined by a sequence $\alpha = \{a_j\}$ .

Determinant Formula: $\det(A_R(n)) = (\prod a_j) [1 + \sum_{k=2}^n \frac{\mu(k)}{a_k}]$ .
Singular Cases: Unlike the Fibonacci case, general sequences can yield singular matrices or matrices with multiple eigenvalues (e.g., double eigenvalues) and complex conjugate pairs depending on the sequence parameters.
Examples: The authors analyze sequences like $a_j = j^p$ and $a_j = j/\ln^2 j$ , deriving specific asymptotic behaviors for their determinants involving the Riemann zeta function and Euler-Mascheroni constant.

4. Significance

Structural Insight: The paper demonstrates how replacing constant entries with a rapidly growing sequence (Fibonacci) fundamentally alters the spectral structure. While the classical Redheffer matrix is linked to the Riemann Hypothesis and has a complex, non-diagonalizable structure with many zero entries in eigenvectors, the Fibonacci-Redheffer matrix is diagonalizable, has strictly non-zero eigenvectors, and possesses a stable negative determinant.
Methodological Advancement: The derivation of the inverse of the Fibonacci-weighted divisor matrix and the subsequent use of rank-one perturbation theory provide a robust framework for analyzing similar structured matrices.
Asymptotic Convergence: The proof that the weighted sum of the Möbius function over Fibonacci numbers converges to a specific negative constant contributes to the understanding of number-theoretic series involving exponential growth sequences.
Counter-Examples: The generalization section highlights that the "nice" properties of the Fibonacci case (real simple eigenvalues, non-zero eigenvectors) are not universal to all Redheffer-type matrices, providing necessary conditions and counter-examples for future research in spectral graph theory and number theory.

In summary, the paper successfully characterizes a new class of structured matrices, proving that the specific growth rate of the Fibonacci sequence ensures a well-behaved, real, and simple spectrum, distinct from the chaotic spectral properties of the original Redheffer matrix.