Fibonacci Q-Matrix

A Fibonacci Matrix of the form

$\begin{displaymath} {\hbox{\sf M}}=\left[{\matrix{m & 1\cr 1 & 0\cr}}\right]. \end{displaymath}$

(1)

and

are defined as Binet Forms

$\displaystyle U_n$	$\textstyle =$	$\displaystyle mU_{n-1}+U_{n-2} \quad (U_0=0, U_1=1)$	(2)
$\displaystyle V_n$	$\textstyle =$	$\displaystyle mV_{n-1}+V_{n-2} \quad (V_0=2, V_1=m),$	(3)

then

$\displaystyle {\hbox{\sf M}}$	$\textstyle =$	$\displaystyle \left[\begin{array}{cc}U_{n+1} & U_n\\ U_n & U_{n-1}\end{array}\right]$	(4)
$\displaystyle {\hbox{\sf M}}^{-1}$	$\textstyle =$	$\displaystyle {\hbox{\sf M}}-m{\hbox{\sf I}}=\left[\begin{array}{cc}0 & 1\\ 1 & -m\end{array}\right].$	(5)

Defining

$\begin{displaymath} {\hbox{\sf Q}}\equiv \left[{\matrix{F_2 & F_1\cr F_1 & F_0\cr}}\right] = \left[{\matrix{1 & 1\cr 1 & 0\cr}}\right], \end{displaymath}$

(6)

then

$\begin{displaymath} {\hbox{\sf Q}}^n=\left[{\matrix{F_{n+1} & F_n\cr F_n & F_{n-1}\cr}}\right] \end{displaymath}$

(7)

(Honsberger 1985, pp. 106-107).

See also Binet Forms, Fibonacci Number

References

Honsberger, R. ``A Second Look at the Fibonacci and Lucas Numbers.'' Ch. 8 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.