OFFSET
1,6
COMMENTS
We consider an important property of the matrix n X n whose elements are the Fibonacci numbers, because this matrix has only two nonzero eigenvalues of the form lambda1 = x + y*sqrt(z) > 0, lambda2 = x - y*sqrt(z) < 0 for n > 1. The other eigenvalues are all 0 (with multiplicity n-2).
Property:
lambda1 + lambda2 = floor(lambda1) = trace (n X n), where trace (n X n) is defined by the sum of the elements on the main diagonal. This property is checked for a large values of n.
For n > 1, trace(n X n) = 2, 24, 670,... = A174997(n).
We observe that y*sqrt(z) - x tends into 1/sqrt(5) when n is infinite.
We give a formulation of the n X n matrix where F_k is the k-th Fibonacci number, as follows:
n = 1 => [F_0] = [0]
n = 2 => [F_0, F_1] = [0, 1]
[F_2, F_3] [1, 2]
n = 3 => [F_0, F_1, F_2] [0, 1, 1]
[F_3, F_4, F_5] = [2, 3, 5]
[F_6, F_7, F_8] [8, 13, 21]
..........................................
The following table gives the first eigenvalues for n = 2, 3, 4, 5 (the
case n = 1 is not listed in the table).
+---+--------------------------+---------------------------+-----------+
| n | positive eigenvalues | negative eigenvalues |eigenvalues|
| | | |equal to 0 |
+---+--------------------------+---------------------------+-----------+
| 2 | 1 + sqrt(2) | 1 - sqrt(2) | |
| 3 | 12 + 2*sqrt(39) | 12 - 2*sqrt(39) |0 |
| 4 | 335 + 2*sqrt(28129) | 335 - 2*sqrt(28129) |0, 0 |
| 5 | 24552 + 2*sqrt(150705906)| 24552 - 2*sqrt(150705906) |0, 0, 0 |
.......................................................................
REFERENCES
G. H. Golub and C. F. van Loan, Matrix Computations, Johns Hopkins, 1989, p. 336.
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
LINKS
S. L. Basin and Verner E. Hoggatt Jr., A Primer on the Fibonacci Sequence--Part II, Fib. Quart. 1, 61-68, 1963.
J. L. Brenner, June Meeting of the Pacific Northwest Section. 1. Lucas' Matrix, Amer. Math. Monthly 58, 220-221, 1951.
Graham Fisher, The Singularity of Fibonacci Matrices, Mathematical Gazette, Vol. 81 (1997), 295-298.
Eric Weisstein's World of Mathematics, Fibonacci Q-Matrix
MAPLE
with(linalg):with(combinat, fibonacci):
for n from 1 to 9 do:
m:=n*n:T:=array(0..m-1):A:=matrix(n, n, T):
for k from 0 to n^2-1 do :
T[k]:=fibonacci(k):
od:
print(eigenvalues(A)) :
od:
MATHEMATICA
sqMax[n_] := If[n == 1, 1, Times@@Power@@@({#[[1]], #[[2]] - Mod[#[[2]], 2]} & /@ FactorInteger [n])]; a[n_] := If[n==0, {0, 0, 0}, Module[{v = Select[ Eigenvalues[Table[Fibonacci[i*(n+1)+j], {i, 0, n}, {j, 0, n}]], #!=0 &]}, s = Simplify[ Total@v ]/2; p = s^2 - Simplify[Times@@v]; sqr=sqMax[p]; {s, Sqrt[ sqr], p/sqr}]]; Array[a, 10, 0] //Flatten (* Amiram Eldar, Dec 16 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 16 2018
STATUS
approved