Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition

FIBONACCI MATRICES

In this chapter, we will take advantage of the power of matrices to extract new identities and results involving Fibonacci and Lucas numbers; see the Appendix for a short discussion of matrices.

20.1 THE Q-MATRIX

First we will demonstrate a close link between matrices and Fibonacci numbers. To this end, consider the matrix

In 1960, Charles H. King studied this matrix, called the Q-matrix, for his Master's Thesis at then San Jose State College, California [360]. Notice that , where denotes the determinant of the square matrix M. Then

Similarly, .

Clearly, an interesting pattern is emerging. More generally, we have the following intriguing result. We will establish it using PMI.

Theorem 20.1

Let . Then

Proof

Clearly, the result is true when . Now assume ...