On the Norms of Toeplitz Matrices with the Pell, Pell-Lucas and Modified Pell Numbers

On the Norms of Toeplitz Matrices with the Pell, Pell-Lucas and Modified Pell Numbers

This paper deals with the usual Pell, Pell-Lucas and Modified Pell numbers and Toeplitz matrices. First, the Toeplitz matrices whose entries are the usual Pell, Pell-Lucas and Modified Pell numbers are constructed, and then the Frobenius, row and column norms of such matrices are computed. Furthermore, the upper and lower bounds for the spectral norms of these matrices are obtained.

___

  • [1] Vajda, Steven. Fibonacci and Lucas numbers, and the golden section: theory and applications. (Courier Corporation, 1989.Koshy, Thomas, 2014).
  • [2] Koshy, Thomasi. Pell and Pell-Lucas numbers with applications. (New York: Springer, 2014).
  • [3] Horadam, A. F., “Pell identities.” Fibonacci Quarterly 9 (1971): 245-263.
  • [4] Horadam, A. F., “Unique minimal representation of integers by negatively subscripted Pell numbers.” The Fibonacci Quarterly 32 (1994): 202-206.
  • [5] Horadam, A. F., “Applications of modified Pell numbers to representations.” Ulam Quarterly 3(1) (1994): 34-53.
  • [6] Melham, R., “Sums involving Fibonacci and Pell numbers.” Portugaliae Mathematica 56.3 (1999): 309-318.
  • [7] Kilic, Emrah, and Dursun Tasci, “The linear algebra of the Pell matrix.” Bol. Soc. Mat. Mexicana, 11 (2005): 3.
  • [8] Dasdemir, Ahmet, “On the Pell, Pell-Lucas and modified Pell numbers by matrix method.” Applied Mathematical Sciences 5.64 (2011): 3173-3181.
  • [9] Mathias, Roy, “The spectral norm of a nonnegative matrix.” Linear Algebra and its Applications 139 (1990): 269-284.