Fibonacci vector and matrix p-norms

1 Introduction

Within matrix theory, the concept of a norm has emerged as a fundamental tool in the analysis of mathematical structures, providing a rigorous means to quantify the magnitude or distance of algebraic entities such as vectors and matrices. Norms are applied in a wide range of disciplines, including optimization, linear algebra, and machine learning, where they facilitate the characterization of structural and functional properties of mathematical objects [1]. Vector norms, such as the Euclidean norm, measure the length of a vector, while matrix norms generalize these notions to matrices, offering insights into their behavior and structure. Among these, the Frobenius norm, denoted by ∣ ∣ A ∣ ∣ 2 , is one of the most widely utilized due to its computational simplicity and interpretability

The Fibonacci sequence { 1 , 1 , 2 , 3 , 5 , 8 , 13 , … } is defined through the n th Fibonacci number F n and the classical recurrence relation:

Combining matrix theory and Fibonacci numbers leads to examining determinants, norms, and characteristic properties associated with various matrix types. Some recent examples are the study of r -min and r -max matrices populated with special cases of Fibonacci numbers [2,3]. Other studies deal with circulant matrices [4] or tridiagonal matrices and trigonometric identities to propose a method for complex factorization of bi-periodic Fibonacci numbers [5].

This article investigates the norms of vectors and matrices composed of Fibonacci numbers. Building upon previous studies in this area [6–9], we examine their theoretical underpinnings and derive several key properties. The principal contribution is a comprehensive analysis of the behavior of Fibonacci vector and matrix p -norms as the parameter p varies. In particular, we derive a closed-form expression for the value of p that minimizes the difference between the p -norm and the infinity norm, allowing for an arbitrarily small discrepancy.

Furthermore, we introduce a novel class of symmetric k -Fibonacci matrices and propose a reorganization technique that significantly simplifies the computation of their p -norms. To build a k -Fibonacci matrix, we rely on the concept of k -Fibonacci sequence { g ( k ) n } defined as follows [6]:

and for n > k ≥ 2 :

The k -Fibonacci sequence reduces to the Fibonacci sequence when k = 2 . From the k -Fibonacci sequence, an n × n symmetric k -Fibonacci matrix Q ( k ) n = [ q ( k ) i j ] is defined [6]

where g ( k ) i j = 0 for j ≤ 0 .

The definition of matrix Q ( k ) n implies the presence of non- k -Fibonacci numbers. For instance, q ( 2 ) 33 = ∑ l = 1 3 g l 2 = g 1 2 + g 2 2 + g 3 2 = 1 2 + 1 2 + 2 2 = 6 , which is not a Fibonacci number. To solve this limitation, we propose the definition of a new n × n   S -type symmetric k -Fibonacci matrix, denoted by S ( k ) n , to ensure that all matrix elements are k -Fibonacci numbers and each element is the sum of k previous elements in rows and columns. By reorganizing the elements of S ( 2 ) n , we show a triangular structure of Fibonacci numbers with a clear pattern that simplifies the computation of its p -norm. Several other properties are also discussed.

The implications of our findings are twofold. First, exploring norms in the context of Fibonacci-based structures yields theoretical insights and potential applications in number theory. Second, the ability to approximate the infinity norm through an optimally chosen p -norm may enhance algorithmic efficiency in numerical computations. Additionally, introducing symmetric k -Fibonacci matrices provides a new framework for simplifying norm evaluations in structured matrix analysis.

This article is structured as follows. Section 2 provides useful background on Fibonacci numbers and k -Fibonacci number. After a general definition of norm and p -norm, Sections 3 and 4 describe a number of properties involving, respectively, Fibonacci vector and matrix p -norms. Section 5 relies on vector p -norms to introduce additional results on distances between Fibonacci vectors, and Section 6 concludes describing natural extensions of this work.

2 Useful background

The Fibonacci sequence F n is defined as:

F n is called the n th Fibonacci number, and the Fibonacci sequence is

A closed-form expression for the n th Fibonacci number is given by the Binet formula [10] involving the golden ratio φ :

The sum of the first n Fibonacci numbers is [11]

The sum of the squares of the first n Fibonacci numbers is [11]

Finally, the sum of the cubes of the first n Fibonacci numbers is [11]

Generalized k -Fibonacci numbers were studied in [6–8] with alternative definitions. The k -Fibonacci sequence { g ( k ) n } is defined in [6] as

and for n > k ≥ 2 :

that reduces to the Fibonacci sequence when k = 2 . From this k -Fibonacci sequence, two types of Fibonacci matrices were also introduced in [6]. First, an n × n k -Fibonacci matrix ℱ ( k ) n = [ f ( k ) i j ] is defined as

This definition places consecutive k -Fibonacci numbers in a lower triangular matrix form. This matrix is useful to find consecutive k -Fibonacci numbers from the first to the n th k -Fibonacci number. An example of a 2-Fibonacci matrix of order 5 is

Second, an n × n symmetric k -Fibonacci matrix Q ( k ) n = [ q ( k ) i j ] is defined as

where g ( k ) i j = 0 for j ≤ 0 . An example of a 2-Fibonacci symmetric matrix of order 5 is

This definition implies the presence of non- k -Fibonacci numbers such as 6, 9, 15, 24, and 40 in the previous example.

3 Fibonacci vector p -norms

Before discussing Fibonacci vector p -norms, let us first define norms:

A special class of norms is the p -norm, denoted by ∣ ∣ x ∣ ∣ p for any vector x = ( x 1 , x 2 , … , x n ) , and defined as follows:

Consider an n -dimensional Fibonacci vector by extracting the first column vector of matrices ℱ ( 2 ) n or Q ( 2 ) n , denoted by q n = ( F 1 , F 2 , … , F n ) . Then, its p -norm is

In what follows, we derive several properties from basic identities described in Section 2:

1. ∣ ∣ q n ∣ ∣ p > 0 for n ≥ 1 ,

2. ∣ ∣ q n ∣ ∣ 1 = ∑ i = 1 n F i = F n + 2 − 1 ,

3. ∣ ∣ q n ∣ ∣ 2 2 = ∑ i = 1 n F i 2 = F n ⋅ F n + 1 ,

4. ∣ ∣ q n ∣ ∣ 3 3 = ∑ i = 1 n F i 3 = 1 2 ( F n ⋅ F n + 1 2 + ( − 1 ) n ⋅ F n − 1 − 1 ) ,

5. ∣ ∣ q n ∣ ∣ ∞ = lim p → ∞ ∑ i = 1 n F i p 1 ⁄ p = max { F 1 , F 2 , … , F n } = F n ,

6. ∣ ∣ q n ∣ ∣ − ∞ = lim p → − ∞ ∑ i = 1 n F i p 1 ⁄ p = min { F 1 , F 2 , … , F n } = F 1 ,

7. ∣ ∣ q n ∣ ∣ − 1 = ∑ i = 1 n 1 ⁄ F i − 1 = 5 ∑ i = 1 n 1 φ i − ( 1 − φ ) i − 1 ,

8. ∣ ∣ q n ∣ ∣ 0 n = ∏ i = 1 n F i .

By focusing on the value of p that places the difference between the p -norm and the infinite norm below a given threshold, we derive the following result:

From inequality equation (24), two additional properties are derived:

1. ∣ ∣ q n ∣ ∣ p ≥ ∣ ∣ q n ∣ ∣ s ∀ p < s ,

2. ∣ ∣ q n ∣ ∣ p is decreasing in p .

Let us focus now on r -dimensional Fibonacci vectors to define an ( n , r ) -Fibonacci vector as

Along the lines of properties (1)–(10), different values of p lead to the following properties for n , r ≥ 1 :

1. ∣ ∣ q n , r ∣ ∣ p > 0 ,

2. ∣ ∣ q n , r ∣ ∣ 1 = ∣ ∣ q n + r ∣ ∣ 1 − ∣ ∣ q n ∣ ∣ 1 = F n + r + 2 − F n + 2 ,

3. ∣ ∣ q n , r ∣ ∣ 2 2 = ∣ ∣ q n + r ∣ ∣ 2 2 − ∣ ∣ q n ∣ ∣ 2 2 = F n + r ⋅ F n + r + 1 − F n ⋅ F n + 1 ,

4. ∣ ∣ q n , r ∣ ∣ 3 3 = ∣ ∣ q n + r ∣ ∣ 3 3 − ∣ ∣ q n ∣ ∣ 3 3 = 1 2 ( F n + r ⋅ F n + r + 1 2 + ( − 1 ) n + r ⋅ F n + r − 1 − 1 ) − 1 2 ( F n ⋅ F n + 1 2 + ( − 1 ) n ⋅ F n − 1 − 1 ) ,

5. ∣ ∣ q n , r ∣ ∣ − 1 = 5 ∑ i = n + 1 n + r 1 φ i − ( 1 − φ ) i − 1 ,

6. ∣ ∣ q n , r ∣ ∣ ∞ = F n + r ,

7. ∣ ∣ q n , r ∣ ∣ − ∞ = F n + 1 ,

8. ∣ ∣ q n , r ∣ ∣ 0 n = ∏ i = n + 1 n + r F i ,

9. ∣ ∣ q n , r ∣ ∣ p ≥ ∣ ∣ q n , r ∣ ∣ s ∀ p < s ,

10. ∣ ∣ q n , r ∣ ∣ p is decreasing in p .

4 Fibonacci matrix p -norms

Given matrix A of size m × n , let us consider an entry-wise matrix norm defined as follows:

where vec ( A ) is a vectorization function of matrix A that produces a vector v of m ⋅ n elements obtained by arranging the elements of A in row-major order, i.e., by arranging them sequentially row by row. Formally, v n ( i − 1 ) + j = a i j , where a i j is the ( i , j ) -entry of matrix A. Note that when p = 2 , ∣ ∣ A ∣ ∣ 2 is the Frobenius matrix norm.

Consider the 2-Fibonacci matrix of order n as defined in the introduction [6]:

Note that ℱ ( 2 ) n is built from the concatenation of n vectors q i with i ∈ { 1 , … , n } . Then, p -norm of ℱ ( 2 ) n is given by

Different values of p lead to the following properties:

1. ∣ ∣ ℱ ( 2 ) n ∣ ∣ 1 = ∑ i = 1 n ∣ ∣ q i ∣ ∣ 1 = ∑ i = 1 n ( F i + 2 − 1 ) = ∣ ∣ q 2 , n + 2 ∣ ∣ 1 − n ,

2. ∣ ∣ ℱ ( 2 ) n ∣ ∣ 2 2 = ∑ i = 1 n ∣ ∣ q i ∣ ∣ 2 2 = ∑ i = 1 n F i ⋅ F i + 1 ,

3. ∣ ∣ ℱ ( 2 ) n ∣ ∣ 3 3 = ∑ i = 1 n ∣ ∣ q i ∣ ∣ 3 3 = 1 2 ∑ i = 1 n ( F i ⋅ F i + 1 2 + ( − 1 ) i ⋅ F i − 1 − 1 ) ,

4. ∣ ∣ ℱ ( 2 ) n ∣ ∣ ∞ = F n ,

5. ∣ ∣ ℱ ( 2 ) n ∣ ∣ − ∞ = 0 ,

6. ∣ ∣ ℱ ( 2 ) n ∣ ∣ − 1 = ∞ ,

7. ∣ ∣ ℱ ( 2 ) n ∣ ∣ 0 n = 0 .

A limitation of the symmetric k -Fibonacci matrix Q ( k ) n = [ q ( k ) i j ] described in [6] is that it cannot be fully represented by k -Fibonacci numbers. Let us define a new n × n   S -type symmetric k -Fibonacci matrix, denoted by S ( k ) n = [ s ( k ) i j ] , and defined as

This definition ensures that all matrix elements are k -Fibonacci numbers and each element is the sum of k preceding elements in rows and columns. Then, a 2-Fibonacci symmetric matrix of order n is

Reorganizing the elements of S ( 2 ) n leads to a triangular structure with a clear pattern:

An example of this reorganization for the elements of S ( 2 ) 5 is shown in Table 1. Note that the sum of the entries of the i th row in the table is ( 5 − ∣ 5 − i ∣ ) F i , where F i is the i th Fibonacci number. By induction, for matrix S ( 2 ) n of order n , the sum of the elements of the i th row is ( n − ∣ n − i ∣ ) F i . Analogously, the sum of the squared entries of the i th row is ( 5 − ∣ 5 − i ∣ ) F i 2 . And the product of the entries of the i th row equals F i 5 − ∣ 5 − i ∣ . The totals of the last three columns of Table 1 correspond to the norms ∣ ∣ S ( 2 ) 5 ∣ ∣ , ∣ ∣ S ( 2 ) 5 ∣ ∣ 2 2 , ∣ ∣ S ( 2 ) 5 ∣ ∣ 0 5 .

Another example of this reorganization for the elements of S ( 3 ) 6 is shown in Table 2. Note that the structure of the reorganization and the emerging pattern that allows for easy computation of norms is independent of k . In other words, the reorganization in equation (35) is valid for any k -Fibonacci matrix S ( k ) n , provided that at least 2 n − 1 elements of the sequence are defined. This property leads to the following result for the p -norm of any S ( k ) n matrix:

Without loss of generality by assuming that F i represents a general k -Fibonacci number, we set p to different values to derive the following ∣ ∣ S ( k ) n ∣ ∣ p p norm properties:

1. ∣ ∣ S ( k ) n ∣ ∣ 1 = ∑ i = 1 2 n − 1 ( n − ∣ n − i ∣ ) F i ,

2. ∣ ∣ S ( k ) n ∣ ∣ 2 2 = ∑ i = 1 2 n − 1 ( n − ∣ n − i ∣ ) F i 2 ,

3. ∣ ∣ S ( k ) n ∣ ∣ 3 3 = ∑ i = 1 2 n − 1 ( n − ∣ n − i ∣ ) F i 3 ,

4. ∣ ∣ S ( k ) n ∣ ∣ ∞ = F 2 n − 1 ,

5. ∣ ∣ S ( k ) n ∣ ∣ − ∞ = F 1 ,

6. ∣ ∣ S ( k ) n ∣ ∣ − 1 = ∑ i = 1 2 n − 1 ( n − ∣ n − i ∣ ) F i − 1 ,

7. ∣ ∣ S ( k ) n ∣ ∣ 0 n = ∑ i = 1 2 n − 1 F i n − ∣ n − i ∣ .

Summarizing, the main motivation to propose S ( k ) n is to produce a symmetric matrix with the desired property of ensuring that all entries are Fibonacci numbers. In addition to its concise definition, the main implication is the advantage of computing matrix norms using simple closed-form expressions, as shown in Properties 29–34. These expressions stem from a key reorganization step that allows for the detection of clear patterns in the structure of the proposed matrices disregarding the value of k .

5 Distances between vectors and matrices

When considering two vectors (matrices) of the same size, the p -norm of the difference of the two vectors (matrices) implies a p -distance between them. To provide a general p -distance definition that applies to vectors and matrices, it is convenient to transform matrices into vectors through the vectorization operator defined in Section 4. Then, the p -distance between two n -dimensional Fibonacci vectors x and y is defined as:

Consider two ( n , r ) -Fibonacci vectors whose elements are separated by d positions in the sequence x = q n + d , r and y = q n , r . Then

and

From the fundamental identity of Fibonacci numbers F n + 1 − F n − 1 = F n , for d = 2 , we have

Setting d = 3 implies that F n + 4 − F n + 1 = F 3 ⋅ F n + 2 and F n + 3 + r − F n + r = F 3 ⋅ F n + r + 1 , leading to

Consider now the sum and the difference of two ( n , r ) -Fibonacci vectors whose elements are separated by d positions in the sequence:

Then, the addition of the 1-norms of the sum and difference vectors is

Similarly, using the parallelogram law in inner product spaces for 2-norms, we have

By considering the golden ratio,

we can approximate the product of two consecutive Fibonacci numbers through

to further simplify the addition of the 2-norms of the sum and difference of ( n , r ) -Fibonacci vectors whose elements are separated by d positions in the sequence:

6 Conclusions

By defining a p -norm over the set of vectors and matrices populated with Fibonacci numbers, several essential properties are derived by varying parameter p . Given the decreasing character of the difference of p -norms as p increases, a closed-form expression is given to obtain the value of p that makes the difference between the p -norm and the infinite norm as small as needed. A new symmetric k -Fibonacci matrix is defined, and a reorganization of its elements leads to a pattern that simplifies the computation of its p -norms. The analysis of p -distances between same-sized vectors (matrices) is illustrated by specific examples when the elements are separated by two and three positions in the sequence.

The proposed symmetric k -Fibonacci matrix represents an extension of the work by [6] ensuring that all entries are k -Fibonacci numbers. Connecting with the main topic of this article, the advantage of this new class of matrices is the possibility to compute norms through a closed-form expression derived from a key reorganization of its elements. Acknowledging that the generalization of the results in this article for the S ( k ) class of matrix to the ℱ ( k ) and Q ( k ) classes is not a trivial question because of the different structures derived from their definition, we consider that this research question represents an interesting future line of research.