%I #32 Jan 05 2025 19:51:41
%S 0,0,0,1,1,2,12,2,39,335,2,28129,24552,2,150705906,4778306,2,
%T 5708053098753,2456043408,52,2230824417240678,3322443961836,3,
%U 1226514875504825822765861,11804340768687390,4,8708903811455950754101585556070
%N Greatest eigenvalues of the form x + y*sqrt(z) of the Fibonacci matrix of the form n X n = [[F(0),F(1),...,F(n-1)],[F(n),F(n+1),...,F(2n-1)],...,[F(n(n-1)),F(n(n-1)+1),...,F(n^2-1)]] represented by the triples of integers (x, y, z).
%C 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).
%C Property:
%C 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.
%C For n > 1, trace(n X n) = 2, 24, 670,... = A174997(n).
%C We observe that y*sqrt(z) - x tends into 1/sqrt(5) when n is infinite.
%C We give a formulation of the n X n matrix where F_k is the k-th Fibonacci number, as follows:
%C n = 1 => [F_0] = [0]
%C n = 2 => [F_0, F_1] = [0, 1]
%C [F_2, F_3] [1, 2]
%C n = 3 => [F_0, F_1, F_2] [0, 1, 1]
%C [F_3, F_4, F_5] = [2, 3, 5]
%C [F_6, F_7, F_8] [8, 13, 21]
%C ..........................................
%C The following table gives the first eigenvalues for n = 2, 3, 4, 5 (the
%C case n = 1 is not listed in the table).
%C +---+--------------------------+---------------------------+-----------+
%C | n | positive eigenvalues | negative eigenvalues |eigenvalues|
%C | | | |equal to 0 |
%C +---+--------------------------+---------------------------+-----------+
%C | 2 | 1 + sqrt(2) | 1 - sqrt(2) | |
%C | 3 | 12 + 2*sqrt(39) | 12 - 2*sqrt(39) |0 |
%C | 4 | 335 + 2*sqrt(28129) | 335 - 2*sqrt(28129) |0, 0 |
%C | 5 | 24552 + 2*sqrt(150705906)| 24552 - 2*sqrt(150705906) |0, 0, 0 |
%C .......................................................................
%D G. H. Golub and C. F. van Loan, Matrix Computations, Johns Hopkins, 1989, p. 336.
%D Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%H S. L. Basin and Verner E. Hoggatt Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/1-1/primer-1.pdf">A Primer on the Fibonacci Sequence--Part II</a>, Fib. Quart. 1, 61-68, 1963.
%H J. L. Brenner, <a href="https://www.jstor.org/stable/2306613">June Meeting of the Pacific Northwest Section. 1. Lucas' Matrix</a>, Amer. Math. Monthly 58, 220-221, 1951.
%H Graham Fisher, <a href="http://www.jstor.org/stable/3619219">The Singularity of Fibonacci Matrices</a>, Mathematical Gazette, Vol. 81 (1997), 295-298.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciQ-Matrix.html">Fibonacci Q-Matrix</a>
%p with(linalg):with(combinat,fibonacci):
%p for n from 1 to 9 do:
%p m:=n*n:T:=array(0..m-1):A:=matrix(n,n,T):
%p for k from 0 to n^2-1 do :
%p T[k]:=fibonacci(k):
%p od:
%p print(eigenvalues(A)) :
%p od:
%t 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 *)
%Y Cf. A000045, A174997.
%K nonn
%O 1,6
%A _Michel Lagneau_, Dec 16 2018