🔢 mathematics

A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity

This paper establishes a universal identity for powers of elements in quadratic algebras to derive a general formula for 2x2 matrix powers based on trace and determinant, which is then applied to the Fibonacci matrix to prove a binomial expansion formula for $F_{nm}$ , demonstrating that such identities stem from general algebraic principles rather than specific Fibonacci properties.

Original authors: Marco Mantovanelli

Published 2026-03-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Marco Mantovanelli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 trying to predict the future of a number, but you only have two rules to work with: a "step forward" rule and a "step back" rule. In the world of mathematics, there's a special club called Quadratic Algebras where every number (or matrix) follows a simple quadratic rule: if you square it, it can be rewritten as a mix of itself and a constant.

This paper by Marco Mantovanelli is like discovering a universal remote control for these numbers. Here is the breakdown of what he found, explained through everyday analogies.

1. The Problem: The "Exponential Explosion"

Usually, when you take a number $x$ and raise it to a high power (like $x^{100}$ ), the result gets incredibly complicated. It looks like a messy tower of blocks.

However, in this specific mathematical club, there's a secret shortcut. Because every member follows the rule $x^2 = tx - d$ (where $t$ and $d$ are just two special numbers attached to $x$ ), you don't need a messy tower to describe $x^{100}$ . You can actually flatten the whole thing down into a simple line: $x^{100} = A \cdot x + B$ .

The author proves a Universal Identity: No matter what $x$ is, as long as it follows that quadratic rule, its high powers can always be reduced to a simple formula involving just $x$ and the identity (the number 1).

2. The Magic Formula: The "Recipe"

The paper provides a specific recipe (a polynomial) to calculate those coefficients $A$ and $B$ .

Think of $x$ as a character in a story.
Think of $t$ (the trace) and $d$ (the determinant) as the character's personality traits.
The author shows that if you know the character's personality traits, you can predict exactly what the character will look like after $m$ chapters (powers), without reading the whole book.

The formula uses a pattern similar to Pascal's Triangle (binomial coefficients), which is the same math used to calculate odds in gambling or how many ways you can arrange items in a line.

3. The Matrix Connection: The "2x2 Machine"

The paper then applies this to 2x2 matrices (grids of four numbers).

Imagine a matrix as a machine that transforms shapes.
The "Trace" is the machine's total energy, and the "Determinant" is its scaling factor.
The famous Cayley-Hamilton Theorem says this machine obeys the same quadratic rule mentioned above.
The Breakthrough: The author shows that you can calculate the result of running this machine $m$ times ( $M^m$ ) using only its energy (Trace) and scaling factor (Determinant). You don't need to do the messy multiplication $m$ times. You just plug the energy and scaling factor into the "Universal Recipe."

4. The Fibonacci Connection: Solving a Famous Puzzle

Finally, the paper connects this to the Fibonacci numbers (1, 1, 2, 3, 5, 8...), which appear everywhere in nature, from sunflower seeds to spiral galaxies.

Fibonacci numbers are generated by a specific 2x2 matrix machine.
Mathematicians have long struggled to find a simple formula for $F_{nm}$ (the $nm$-th Fibonacci number) based on $L_n$ (Lucas numbers, a cousin of Fibonacci).
A recent mathematician named Vorobtsov found a complex formula for this.
The Paper's "Aha!" Moment: Mantovanelli shows that Vorobtsov's formula isn't a magical coincidence specific to Fibonacci numbers. It is simply a special case of the Universal Identity.
By plugging the Fibonacci machine's "personality traits" (Trace and Determinant) into the Universal Recipe, the Fibonacci formula pops out automatically.

The Big Takeaway

The paper argues that we often get too obsessed with the specific details of famous number sequences like Fibonacci. Instead, we should look at the underlying structure.

The Analogy:
Imagine you are trying to predict the weather in three different cities (New York, London, Tokyo).

Old Way: You study the specific history of rain in each city separately and write three different, complex books.
This Paper's Way: You realize all three cities follow the same basic laws of physics (pressure, temperature, humidity). You write one single book on atmospheric physics. Now, you can predict the weather in any city just by plugging in that city's current pressure and temperature.

In short: The paper proves that complex identities involving Fibonacci numbers are just the result of a simple, universal algebraic law that applies to any system governed by a quadratic rule. It turns a "magic trick" into a predictable, mechanical process.

1. Problem Statement

The paper addresses the structural derivation of identities involving powers of elements in quadratic algebras, specifically focusing on the expansion of $x^m$ where $x$ satisfies a quadratic relation $x^2 - tx + d = 0$ .

While specific identities for Fibonacci ( $F_n$ ) and Lucas ( $L_n$ ) numbers are well-known, recent work by Vorobtsov provided an explicit binomial expansion for $F_{nm}$ in terms of $L_n$ . The author seeks to demonstrate that such identities are not merely coincidental properties of Fibonacci numbers but are consequences of a universal algebraic principle governing any element in a quadratic algebra (or $2 \times 2$ matrices) defined by its trace and determinant.

2. Methodology

The author employs a combination of algebraic induction, polynomial recurrence relations, and matrix theory (specifically the Cayley–Hamilton theorem).

Quadratic Reduction: The core method involves assuming an element $x$ in an $R$ -algebra satisfies $x^2 = tx - d$ . The author proves by induction that any power $x^m$ can be reduced to a linear combination of $x$ and the identity element $1$:
$x^m = a_m x + b_m$
By multiplying by $x$ and substituting the quadratic relation, recurrence relations for the coefficients $a_m$ and $b_m$ are derived.
Polynomial Definition: The coefficients are identified as specific polynomials $P_m(t, d)$ defined by the recurrence:
$P_{m+1}(t, d) = t P_m(t, d) - d P_{m-1}(t, d)$
with initial conditions $P_0 = 0$ and $P_1 = 1$ .
Matrix Application: The theory is applied to the ring of $2 \times 2$ matrices $M_2(R)$ . By the Cayley–Hamilton theorem, any matrix $M$ satisfies its characteristic equation $M^2 - \text{tr}(M)M + \det(M)I = 0$ . Thus, the universal identity applies directly to matrix powers.
Fibonacci Specialization: The Fibonacci matrix $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ is used as a specific instance. The author calculates the trace and determinant of $A^n$ (which are $L_n$ and $(-1)^n$ , respectively) and applies the general matrix formula to the $(1,2)$ entry to isolate $F_{nm}$ .

3. Key Contributions

A. Universal Quadratic Identity (Theorem 1)

The paper establishes a general formula for powers of elements in quadratic algebras. For an element $x$ satisfying $x^2 - tx + d = 0$ :
$x^m = P_m(t, d)x - d P_{m-1}(t, d)$
where $P_m(t, d)$ is explicitly given by the binomial sum:
$P_m(t, d) = \sum_{i=0}^{\lfloor (m-1)/2 \rfloor} \binom{m-1-i}{i} t^{m-1-2i} (-d)^i$
This result is universal, holding for any commutative ring $R$ and any $R$ -algebra.

B. Matrix Power Formula (Corollary 1)

A direct consequence for $2 \times 2$ matrices $M$ with trace $t$ and determinant $d$ is:
$M^m = P_m(t, d)M - d P_{m-1}(t, d)I$
This provides a closed-form expression for matrix powers dependent solely on trace and determinant, independent of the specific matrix entries.

C. Connection to Chebyshev Polynomials

The author notes that $P_m(t, d)$ is a scaled version of Chebyshev polynomials of the second kind ( $U_n$ ). If $\sqrt{d}$ exists, the identity generalizes the classical representation:
$P_m(t, d) = d^{(m-1)/2} U_{m-1}\left(\frac{t}{2\sqrt{d}}\right)$

D. Derivation of Vorobtsov's Identity (Corollary 2)

By applying the matrix formula to the Fibonacci matrix $A^n$ , the paper derives a binomial expansion for $F_{nm}$ :
$F_{nm} = F_n \sum_{i=0}^{\lfloor (m-1)/2 \rfloor} \binom{m-1-i}{i} L_n^{m-1-2i} (-1)^{i(n+1)}$
This recovers a recent result by Vorobtsov but provides a more direct, structural derivation.

4. Results

Generalization: The paper successfully generalizes specific Fibonacci identities to a universal framework applicable to any quadratic algebra.
Explicit Formula: It provides a concrete, computable binomial sum for the coefficients of matrix powers, avoiding the need for diagonalization or eigenvalue decomposition.
Unification: It unifies the theory of second-order linear recurrences, matrix powers, and Chebyshev polynomials under a single algebraic reduction principle.

5. Significance

Conceptual Clarity: The work shifts the perspective from viewing Fibonacci identities as number-theoretic curiosities to viewing them as inevitable consequences of the Cayley–Hamilton theorem and quadratic algebra structures.
Simplification: It offers a streamlined method for deriving complex identities involving $F_{nm}$ and $L_n$ without resorting to ad-hoc combinatorial proofs.
Broad Applicability: The "Universal Quadratic Reduction" is applicable beyond Fibonacci numbers to any system governed by a second-order recurrence or $2 \times 2$ matrix dynamics, including Lucas numbers, Pell numbers, and other linear recurrence sequences.

In conclusion, Mantovanelli demonstrates that the complex binomial structures found in Fibonacci identities are simply the manifestation of a fundamental algebraic reduction process inherent to quadratic relations.