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On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers. (English) Zbl 1204.15031

Let \(\{t_n \}_{n = -\infty}^{\infty}\) be a double infinite sequence. A Toeplitz matrix is an \(n \times n\) matrix \[ T_n = \left [ \begin{matrix} t_0 & t_{-1} & t_{-2} & ... & t_{-(n-1)} \\ t_1 & t_0 & t_{-1} & ... & t_{-(n-2)} \\ t_2 & t_1 & t_0 & ... & t_{-(n-3)} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ t_{-(n-1)} & t_{-(n-2)} & t_{-(n-3)} & ... & t_0 \end{matrix} \right ]. \] The spectral norm of the matrix \(A\) is \[ \| A\|_2 = \sqrt{\max_{1 \leq i \leq n} |\lambda_i|}, \] where the numbers \(\lambda_i\) are the eigenvalues of the matrix \(A^HA\) and the matrix \(A^H\) is the conjugate transpose of the matrix \(A\).
Theorem 1: Let the \(n \times n\) matrix \(A = [a_{i,j}]\) satisfy \(a_{i,j} = F_{i-j}\), where \(\{F_n\}_{n = -\infty}^{\infty}\) is the double infinite Fibonaccci sequence, for which \(F_0 = 0, F_1, F_n = F_{n-1} + F_{n-2}, F_{-n} =(-1)^{n+1}F_n\) for \(n \geq 2\). Then \[ \sqrt{(1+F_nF_{n+1})F_nF_{n-1}} \geq \| A\|_2 \geq \begin{cases} \sqrt{\frac{2}{n} F_n^2}, &\text{if \(n\) is even} \\ \sqrt{\frac{2}{n} (F_n^2 - 1)}, &\text{if \(n\) is odd.} \end{cases} \]
Theorem 2: Let the \(n \times n\) matrix \(B = [b_{i,j}]\) satisfy \(b_{i,j} = L_{i-j}\), where \(\{L_n\}_{n = -\infty}^{\infty}\) is the double infinite Lucas sequence, for which \(L_0 = 2, L_1, L_n = L_{n-1} + L_{n-2}, L_{-n} = (-1)^{n}L_n\) for \(n \geq 2\). Then \[ \sqrt{(L_nL_{n-1}-1)(L_nL_{n-1} + 2)} \geq \| B\|_2 \geq \begin{cases} \sqrt{\frac{2}{n} (L_n^2 - 4)}, &\text{if \(n\) is even} \\ \sqrt{\frac{2}{n} (L_n^2 + 1)}, &\text{if \(n\) is odd.} \end{cases} \]

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B36 Matrices of integers
11B39 Fibonacci and Lucas numbers and polynomials and generalizations