Fibonacci Q-matrix

In the last decades the theory of Fibonacci numbers was supplemented by the theory of so-called Fibonacci Q-matrix. The latter presents itself the simplest 2 ´ 2 matrix of the following form:

 (1)

Note that the determinant of the Q-matrix is equal -1.

In the paper of the famous American Fibonacci-mathematician H.W. Gould "A history of the Fibonacci Q-matrix and a higher-dimensional problem" ("The Fibonacci Quarterly", 1981, 19) devoted to the memory of Verner E. Hoggat, the creator of the Fibonacci Association, it was stated the history of the Q-matrix, given the extensive bibliography on the Q-matrix and on the related questions and emphasized the Hoggatt's contribution in development of the Q-matrix theory. Although the name of the "Q-matrix" was introduced before Verner E. Hoggat, just from the paper by Basin&Hoggatt (1963) the idea of the Q-matrix "caught on like wildfire among Fibonacci enthusiasts. Numerous papers have appeared in Fibonacci Quarterly authored by Hoggatt and/or his students and other collaborators where the Q-matrix method became a central tool in the analysis of Fibonacci properties".

But what relation has the Q-matrix to Fibonacci numbers? To answer this question it is necessary to take the n-th power of the Q-matrix. Then we will get:

 (2)

where Fn-1, Fn, Fn+1 are the Fibonacci numbers.

But we know that Det (An) = (Det A)n. It follows from this the following property for the determinant of the Q-matrix:

 Det Qn = (-1)n, (3)

where n is an integer.

But if we calculate Det Qn using (2) and use (3) then we get the following identity connecting three neighboring Fibonacci numbers:

 (4)

Thus, this means that the Q-matrix express one of the most important properties of Fibonacci numbers given with (4)!

Let's represent now the matrix (2) in the following form:

 (5)

or

 Qn = Qn-1 + Qn-2. (6)

Let's write the expression (6) in the following form:

 Qn-2 = Qn - Qn-1. (7)

The explicit forms of the matrices Qn (n = 0, ±1, ±2, ±3, ...) obtained by means of the recurrent formulas of (6), (7) are given in Table 1.

Table 1.

 n 0 1 2 3 4 5 6 7 Qn Q -n

Note that Table 1 gives the "direct" matrices Qn and their "inverse" matrices Q -n in the explicit form. Comparing the "direct" and "inverse" Fibonacci matrices Qn and Q -n given in Table 1 it is easy to see that there exists a very simple method to get the "inverse" matrix Q -n from its "direct" matrix Qn. Really, if the power of the "direct" matrix Qn given with (2) is the even number (n = 2k) then for obtaining of its "inverse" matrix Q -n it is necessary to rearrange in the matrix (2) its diagonal entries Fn+1 and Fn-1 and to take its diagonal entries Fn with the opposite sign. It means that for the case n = 2k the "inverse" matrix Q -n has the following form:

 (8)

For the case n = 2k + 1 for obtaining of the "inverse" matrix Q -n from the "direct" matrix Qn it is necessary to rearrange in (2) the diagonal entries Fn+1 and Fn-1 and to take them with the opposite sign, that is:

 (9)

Other method to get the matrix Qn follows directly from the expression of (2). For that it is necessary to present two sequences of Fibonacci numbers shifted one to another in one column (Table 2).

Table 1.

 n 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 Fn+1 21 13 8 5 3 2 1 1 0 1 -1 2 -3 5 -8 Fn 13 8 5 3 2 1 1 0 1 -1 2 -3 5 -8 -13

If we select number n = 1 in the first row of Table 2 and then four Fibonacci numbers in two lower rows (under the numbers of 1 and 0 of the first rows) we can see that a totality of the four Fibonacci numbers forms the Q-matrix. Moving along Table 2 to the left about Q-matrix we will get consecutively the matrices Q2, Q3, ..., Qn. Moving to the right about Q-matrix we will get consecutively the matrices Q0, Q -1, ..., Q -n. As example we can see in Table 2 the matrix Q5 and the "inverse" matrix Q -5 which are singled out with fatty print.

Thus, the studying the Q-matrix is really fascinating pastime and one can understand the enthusiasm of Verner Hoggatt and other Fibonacci mathematicians, which four decades ago begun to study Q-matrix! But we will try to introduce so-called generalized Fibonacci matrices based on the p-Fibonacci numbers resulting from Pascal Triangle. And for that we need to remember and to develop some interesting properties of the p-Fibonacci numbers. And we will do this at the next page of our Museum. Follow us!