Generating Matrices and Fibonacci-Like Numbers

Article History: Received: 11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021 Abstract: Among several classes of interesting numbers, Fibonacci numbers plays a significant role and these numbers very often turns up in several branches of Science, Engineering and Technology. Considering two by two square matrices whose entries are Fibonacci and Fibonacci like sequence numbers, we investigate the integral powers of such matrices and prove interesting results concerning them. Finally, the limiting cases of entries of nth power of such matrices are derived.


Introduction
The concept of Fibonacci Numbers was first introduced to Europe through the memorable book "Liber Abaci" written by Italian mathematician Leonardo Fibonacci. Ever since its publication in 1202 CE, this classic book was responsible for two important mathematical achievements. The first being spreading the modern Hindu -Arabic numeral system throughout the globe thereby replacing the then existing Roman numeral system. Second, in one of many amusing problems present in the book, Fibonacci posed a problem about the growth of immortal rabbits whose solution turns to be Fibonacci Numbers. So much about Fibonacci numbers has been written and investigated since that time that a separate journal in the name "Fibonacci Quarterly" was devoted to study their properties. This journal beginning its journey from 1963 continues to this day to publish new results about these ever fascinating numbers. In this paper, we shall discuss novel approach of generating Fibonacci numbers through square matrices and study the limiting behavior of nth power of such matrices.

Matrices
Let G be a 2 × 2 square matrix be defined by 01

 =  
where G is the matrix defined in (3.1) and n G is product of G performed n times under matrix multiplication.

Proof:
We prove by using Mathematical Induction on number of times that G is multiplied namely n. If n = 1, then equation (4.1) exactly coincide with the matrix provided in definition (3.2). If n = 2, then which matches with (4.1). Thus the result is true for n = 1, 2. By Induction Hypothesis, we now assume that the result is true up to n = k. We now try to prove for n = k + 1.
where k G is a matrix obtained because of Induction Hypothesis for n = k. By equation (2.1), we find that Thus the result is true for n = k + 1 also. Hence by Induction Principle, the result is true for all natural numbers n. This completes the proof.

Theorem 2
For any natural number n, If n = 2, then 12 23 01 1 1 2 which is true by (4.2). Thus the result is true for n = 1, 2. By Induction Hypothesis, we now assume that the result is true up to n = k. We now try to prove for n = k + 1.  Thus the result is true for n = k + 1 also. Hence by Induction Principle, the result is true for all natural numbers n. This completes the proof.
In view of theorems 1 and 2, we see that the matrices G and M are generating matrices for Fibonacci and Fibonacci-Like numbers respectively.

Cassini's Identities
We now prove important identities known as Cassini's Identities for Fibonacci and Fibonaccci-Like numbers using the generating matrices G and M. This completes the proof.

Limiting Matrices
We now prove a basic result from which we can deduce some interesting consequences.

Golden Ratio
The real number known as Golden Ratio denoted by  is defined to be the positive real root of the equation . This proves (a).
This completes the proof of the theorem.
In view of theorem 5, we call the matrices

Conclusion
Using the generating matrices G and M for Fibonacci and Fibonacci-Like numbers, we have proved five interesting theorems in this paper. Theorems 1 and 2 have been proved using mathematical induction. Using these results, we have proved Cassini's Identities for both Fibonacci and Fibonacci-Like numbers in theorem 3. In theorem 4, two identities concerning expressing the powers of generating matrices as linear combination of G and the unit matrix I have been derived. We note that the scalars in such linear combinations are nth and (n-1)th Fibonacci and Fibonacci-Like numbers respectively. Using the two identities of theorem 4, we finally proved in theorem 5, that the limiting matrices produce same matrix whose entries depend only on 1 and the Golden Ratio  . This result is especially amusing in the sense that no matter with what two numbers a, b we begin, if we follow Fibonacci type recurrence relation, then in the limiting case we are sure enough to end with the matrices as shown in equations (6.4) and (6.5) of theorem 5. Using other particular entries of Fibonacci numbers we can similarly, get few more useful results leading to Golden Ratio entries in the final matrices.