Powers and trace of symmetric powers of $2\times 2$ matrices and combinatorial, Fibonacci and Lucas identities

Abstract: Let $A$ be an arbitrary $2\times 2$ matrix. In \cite{Cisneros:PhD,Cisneros:I2x2M} I gave a formula for the trace of the $k$-th symmetric power of $A$ in terms of the anti-diagonal entries of $A<sup>{k+1}$ and $A$. This was based on formulae that I found for the entries of the $k$-th power $A<sup>k$ of the matrix $A$ in terms of its entries but I only sketched the idea of how I obtained such formulae. In this article I give the full proof of those formulae by counting some walks of length $k$ over the complete digraph of order $2$. I compare them with formulae for $A<sup>k$ given by Mc Laughlin in \cite{McLaughlin:CIDnP2x2M} and by Williams in \cite{Williams:nthP2x2M}. This leads to combinatorial identities, in particular expressions for Fibonacci and Lucas numbers.

• The paper derives explicit closed-form expressions for the entries of A^n, linking matrix powers to combinatorial path enumeration.
• It rigorously compares novel formulas with classical results, showing their algebraic equivalence via trace and symmetry methods.
• The paper uncovers connections between symmetric power traces and number sequences, yielding new Fibonacci and Lucas identities with practical computational implications.

Powers and Symmetric Powers of $2\times 2$ Matrices: Trace Formulas and Combinatorial Connections

Introduction and Motivation

This work provides a thorough algebraic and combinatorial analysis of the powers and symmetric powers of arbitrary $2\times2$ complex matrices. It develops explicit formulae for individual entries of $A^n$ in terms of the entries of $A$, ultimately establishing a profound connection between these matrix powers and classical combinatorial sequences, notably the Fibonacci and Lucas numbers. Leveraging the combinatorics of paths in digraphs, the article rigorously proves results that previously appeared only as sketches or folklore, systematically comparing distinct formulas from the literature and using this framework to derive new and classical combinatorial identities.

Explicit Formulas for $A^n$ and Combinatorial Interpretations

The main technical achievement is the derivation of explicit closed-form expressions for all entries of $A^n$ for arbitrary $A={a}{b}{c}{d}$, with

$$A^n = \begin{pmatrix} a_n & b_n \ c_n & d_n \end{pmatrix}.$$

Each entry, such as $a_n$, is expressed as a double sum involving binomial coefficients: $$a_n = a^n + \sum_{s=1}^{\lfloor n/2\rfloor} \sum_{m=0}^{n-2s} \binom{n-s-m}{s}\binom{m+s-1}{m} a^{n-2s-m} b^s c^s d^m,$$ analogous expressions hold for $2\times2$0, $2\times2$1, and $2\times2$2, with their own combinatorial structures.

The derivation is based on interpreting the recursion relations among the entries of $2\times2$3 as walks of length $2\times2$4 on the complete digraph of order 2. Each term in the sum corresponds to walks with specified visitation numbers to loops and arcs, with coefficients tracing the count of such walks. This interpretation underpins the subsequent combinatorial identities and illuminates the structural symmetries between $2\times2$5 and $2\times2$6.

Comparison with Alternative Formulae

The paper meticulously compares these novel explicit expressions with classical results such as:

• Mc Laughlin's formula [McLaughlin 2004]: Expresses $2\times2$7 via a generalized binomial sum $2\times2$8, involving the trace and determinant:

$2\times2$9

such that

$A^n$0

• Williams' eigenvalue formula: Uses the eigenvalues $A^n$1 of $A^n$2 to directly express $A^n$3 in terms of $A^n$4 and the identity, with coefficients reflecting the structure of symmetric polynomials in $A^n$5.

The article clarifies that all these formulations are algebraically equivalent due to the Cayley-Hamilton theorem and elementary properties of $A^n$6 matrices, and it explicitly demonstrates the transformations relating the formulas.

Trace of Symmetric Powers and Invariance Properties

A central insight is the identification and exploitation of the $A^n$7-th symmetric power action, $A^n$8, on homogeneous degree-$A^n$9 polynomials, and in particular, computation of $A$0 via both combinatorial means and in terms of certain anti-diagonal entry ratios in powers of $A$1: $A$2 This quotient is shown to be invariant under conjugation of $A$3, a property not generally shared by other entry-based ratios, and furnishes an explicit link between matrix theory and symmetric function theory.

Furthermore, the entries of $A$4 are reconstructed from the traces of the symmetric powers: $A$5 These relationships facilitate the derivation of a host of combinatorial and number-theoretic identities.

Combinatorial, Fibonacci, and Lucas Identities

From the established formulas, a wealth of combinatorial identities are extracted, many involving binomial coefficients and path-counting arguments, such as: $A$6 Moreover, identities connecting convolutions of binomial coefficients to the Fibonacci and Lucas sequences are obtained by specializing $A$7 to appropriate companion matrices. For instance, the trace formula yields: $A$8 with explicit sum expressions matching classical forms and Binet’s formula, but now as consequences of symmetric representation theory.

The invariant property of the trace of symmetric powers under conjugation of $A$9 leads to infinite families of expressions for $A^n$0 parameterized by invertible changes of basis.

Analogous formulas are developed for Lucas numbers and for variations of the Fibonacci sequence involving even indices, incorporating entries with powers of $A^n$1 or $A^n$2, depending on the specific matrix realization.

Theoretical and Practical Implications

This unified framework interlinks combinatorial path enumeration, linear algebraic powers, symmetric tensor representation theory, and number theory. The explicit forms not only clarify the algebraic structure underpinning classical sequences like Fibonacci and Lucas but also provide algorithmically efficient means to compute high powers of $A^n$3 matrices. The combinatorial insights enhance the understanding of walk enumeration on small graphs and could extend to transfer-matrix methods and spectral techniques in combinatorics.

The symmetry observed in the trace of symmetric powers suggests possible extensions to higher-dimensional analogues, though the combinatorial complexity scales rapidly for $A^n$4.

On a practical level, these results have direct implications for computational mathematics, algorithms requiring fast computation of matrix powers, and symbolic computation in algebraic combinatorics.

Conclusion

The paper rigorously establishes and proves explicit formulas for the entries and symmetric power traces of arbitrary $A^n$5 matrices, situating these results within a strong combinatorial and representation-theoretic context. The equivalence and comparison with classic formulas from Mc Laughlin and Williams are clarified, and the transfer of these results to combinatorial identities—including many for Fibonacci and Lucas numbers—is thoroughly developed. These contributions furnish a comprehensive toolkit for researchers working at the interface of linear algebra, combinatorics, and number theory, and open avenues for future exploration in higher rank settings and other combinatorial matrix analytic contexts.

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Reference: "Powers and trace of symmetric powers of $A^n$6 matrices and combinatorial, Fibonacci and Lucas identities" (2607.05589)