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Fibonacci Q-Matrix

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The Fibonacci Q-matrix is the matrix defined by

Q=[F_2 F_1; F_1 F_0]=[1 1; 1 0],
(1)

where F_n is a Fibonacci number. Then

Q^n=[F_(n+1) F_n; F_n F_(n-1)]
(2)

(Honsberger 1985, p. 106). It was first used by Brenner (Brenner 1951, Hoggatt 1968), and its basic properties were enumerated by King (1960).

The Q-matrices immediately give a number of important Fibonacci identities, including

|Q^n|=|Q|^n,
(3)

which gives

F_(n-1)F_(n+1)-F_n^2=(-1)^n,
(4)
Q^(n+1)Q^n=Q^(2n+1),
(5)

which gives

[F_(n+2) F_(n+1); F_(n+1) F_n][F_(n+1) F_n; F_n F_(n-1)]=[F_(2n+2) F_(2n+1); F_(2n+1) F_(2n)],
(6)

and

Q^mQ^(n-1)=Q^(n+m-1),
(7)

which gives

[F_(m+1) F_m; F_m F_(m-1)][F_n F_(n-1); F_(n-1) F_(n-2)]=[F_(m+n) F_(m+n-1); F_(m+n-1) F_(m+n-2)]
(8)

(Honsberger 1985, pp. 105-106).

See also

Fibonacci Number

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References

Basin, S. L. and Hoggatt, V. E. Jr. "A Primer on the Fibonacci Sequence--Part II." Fib. Quart. 1, 61-68, 1963.Brenner, J. L. "June Meeting of the Pacific Northwest Section. 1. Lucas' Matrix." Amer. Math. Monthly 58, 220-221, 1951.Hoggatt, V. E. Jr. "Belated Acknowledgement." Fib. Quart. 6, 85, 1968.Honsberger, R. "The Matrix Q." §8.4 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 106-107, 1985.King, C. H. "Some Further Properties of the Fibonacci Numbers." Master's thesis. San Jose, CA: San Jose State, 1960.

Referenced on Wolfram|Alpha

Fibonacci Q-Matrix

Cite this as:

Weisstein, Eric W. "Fibonacci Q-Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FibonacciQ-Matrix.html

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