Generalized Fibonacci matrices

One can use the idea of the Fibonacci Q-matrix for obtaining the general Q-matrix for the p-Fibonacci numbers. Let's introduce now the following definition for the Qp-matrix:

 (1)

where the index of p takes the following values: 0, 1, 2, 3, ... .

Note that the Qp-matrix is the square (p + 1) ´ (p + 1)-matrix. It contains the p ´ p unit matrix bordered by the last row of 0's and the first column, which consists of 0's bordered by 1's. For p = 0, 1, 2, 3, 4 the Qp-matrices have the following forms, respectively:

Let's compare the neighboring matrices Q4 and Q3. It is easy to see that the matrix Q4 is reduced to the matrix of Q3 if we cross out in the matrix Q4 the last (5th) column and the next to the last (4s) row. Note that we have 1 on the crossing out of the 5th column and 4s row. Because the sum 5 + 4 is equal to the odd number of 9 it means that determinant of the matrix of Q3 differs from the determinant of the matrix of Q3 only by the sign, that is,

 Det Q4 = - Det Q3. (2)

You should believe us that the result (2) is valid and that the latter follows from the matrix theory!

By analogy it is easy to prove the following correlations for determinants of the neighboring Fibonacci Qp-matrices:

Det Q3 = - Det Q2;     Det Q2 = - Det Q1.

Taking in consideration that Det Q0 = 1 and Det Q1 = -1 we get the following unique mathematical property of the Qp-matrices in the general case.

 Det Qp = (-1)p. (3)

Thus the determinant of each matrix (1) depends on the value of the index p. If the index p is even then the Det Qp = 1 for all matrices of the kind (1). In the opposite case (p is the odd number) Det Qp = 1. Even if you are no mathematician by profession we can assess properly this wonderful mathematical result!

Let's consider now the matrix being the n-th power of the Qp-matrix. We would not tire you with the complicated and "fine" mathematical reasons and give the final result:

 (4)

Thus, the matrix is expressed through p-Fibonacci numbers resulting from Pascal Triangle! And the result (4) is the new secret of the Pascal Triangle!

And now we will try to calculate the determinant of the matrix (4). At first glance this problem is extraordinary complicated. But it seems to be complicated only for those who do not know the matrix theory. Really, it follows from the matrix theory that

 Det = (Det Qp)n. (5)

Using (3) we can write the expression (5) in the form:

 Det Qp = (-1)pn, (6)

where p = 0, 1, 2, 3, ... ; n = 0, ±1, ±2, ±3, ... .

And now we can express our enthusiasm regarding to the result (6) and regarding to the power of the mathematical theories! Really, it is impossible to image that the p-Fibonacci numbers resulting from Pascal Triangle can become the basis of the new and infinite class of the square matrices expressed by (1) and (4). And the result (6) seems to us absolutely incredible! It is impossible to image that the determinant of the matrix (4) is equal always to 1 or to (-1) that follows from (6)!

It is clear that the expressions (4) and (6) give unlimited opportunities for the "Fibonacci investigations" because they allow obtaining the infinite number of the fundamental correlations connecting the p-Fibonacci numbers Fp(n). For example for the 2-Fibonacci numbers (p = 2) we have the following correlation connecting the neighboring 2-Fibonacci numbers:

Det = F2(n + 1)[F2(n - 2)F2(n - 2) - F2(n - 1)F2(n - 3)] +

+F2(n)[F2(n)F2(n - 3) - F2(n - 1)F2(n - 2)] +

+ F2(n - 1)[F2(n - 1)F2(n - 1) - F2(n)F2(n-2)] = 1.

We cannot predict now the role of the -matrices given with (4) and their applications in different branches of mathematics, physics and other sciences. However we believe that this result can become so fundamental as Pascal Triangle generating p-Fibonacci numbers and -matrices (4)!

And at the next pages of our Museum we will try to show how to create the new coding theory based on the -matrices. Follow us!