Determinants and inverses of circulant matrices with complex Fibonacci numbers

Fibonacci Fervour in Linear Algebra and Quantum Information Theory

“…and showed that such matrices are invertible [1]. Here, F * n stands for the complex Fibonacci number defined as F * n = F n + iF n+1 .…”

“…It is for this reason that the set of all positive semi-definite (in short, positive) matrices is referred to as a cone. The numbers p, a, u, l are simply 1 2 tr(T P ), 1 2 tr(T A), 1 2 tr(T U) and 1 2 tr(T L), respectively. Let ξ be a unit vector in C 2 .…”

“…Let {ξ (j) , 0 ≤ j ≤ 3} be as in Example 3.1. Then one may verify that S 1 = {ξ (0) , ξ (2) , ξ (3) } and S 2 = {ξ (1) , ξ (2) , ξ (3) } are both equiangular in C 2 with common angle π 4 .…”

A Note on Bigaussian Pell and Pell-Lucas Numbers

“…Harman [17] gave an extension of Fibonacci numbers into the complex plane and generalized the methods by Horadam. Aydın [18] defined bicomplex Pell and Pell-Lucas numbers as and where i, j, and ij satisfy the conditions Some researchers have studied some properties of Gaussian numbers [19][20][21]. The applications of Pell and Pell-Lucas numbers in mathematics are undeniably large.…”

A Note Concerning Generalized Complex k-Horadam and Generalized Gaussian k-Horadam Squences

“…As in computation of spline functions, time series analysis, signal and image processing, queueing theory, polynomial and power series computations and many other areas, typical problems modelled by Toeplitz matrices are the numerical solution of certain differential and integral equations [1,2,3]. The literature includes many papers dealing with the special number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, Mersenne, Fermat, Padovan and Perrin [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The books written by Koshy and Vajda collect and classify many results dealing with these number sequences, most of which have been obtained quite recently [16,18,19].…”