On the Norms of <i>r</i>-Hankel Matrices Involving Fibonacci and Lucas Numbers

Abstract

Let us define () r ij A H a = = to be n n × r-Hankel matrix. The entries of matrix A are 2 n i j F F + − = or 2 n i j L F + − = where n F and n L denote the usual Fibonacci and Lucas numbers, respectively. Then, we obtained upper and lower bounds for the spectral norm of matrix A. We compared our bounds with exact value of matrix A's spectral norm. These kinds of matrices have connections with signal and image processing, time series analysis and many other problems.

On the Norms of RFMLR-Circulant Matrices with the Exponential and Trigonometric Functions

“…Especially, some scholars studied the norms of r-circulant matrices, geometric circulant matrices, and r-Hankel and r-Toeplitz matrices with some famous numbers and polynomials. For example, on the spectral norms of circulant matrices, r-circulant matrices, geometric circulant matrices, and r-Hankel and r-Toeplitz matrices with Fibonacci number, Lucas number, generalized Fibonacci and Lucas numbers, and generalized k-Horadam numbers have been studied [2][3][4][5][6][7][8]. We have obtained several results [9,10] of the norms of matrices mentioned above with exponential forms e(k/n) and trigonometric functions cos(kπ/n) and sin(kπ/n).…”