An application of matricial Fibonacci identities to the computation of spectral norms (Q457949)

The authors present the exact value for the spectral norms of Toeplitz matrices constructed from Fibonacci and Lucas numbers.NEWLINENEWLINELet \(A=(a)_{i j}\) be a matrix with rank \(k\) and adjoint \(A^*\), and let the singular values of \(A\) be defined by the non-zero eigenvalues of \(|A|=\left (A^* A\right )^{1/2} \), labelled \(s_1,s_2,\dots,s_k\) with \(s_1\geq s_2\geq \ldots \geq s_k>0\). Then, the Schatten norms of the matrix \(A\) are a family of unitary invariant norms, denoted by \(\| A \|_p\) \((1\leq p \leq \infty)\), withNEWLINENEWLINENEWLINE\[NEWLINE \| A \|_p = \left (\sum_{i=1}^k s_i^p\right )^{1/p},\qquad 1 \leq p <\infty,\qquad \text{and}\qquad \| A \|_\infty =s_1. NEWLINE\]NEWLINENEWLINENEWLINEWhen \(p=2\), the Schatten norm equates to the Frobenius norm, and when \(p=\infty\), the Schatten norm returns the maximal singular value of \(A\), denoted by \(\|A\|\), and referred to as the spectral norm.NEWLINENEWLINEThe authors consider the Frobenius and spectral norms of the family of Fibonacci and Lucas matrices, denoted by \(F\) and \(L\), and defined such thatNEWLINENEWLINENEWLINE\[NEWLINE F=\left (f_{i-j}\right )_{i,\, j=0}^{n-1},\qquad\text{and}\qquad L=\left (l_{i-j}\right )_{i,\, j=0}^{n-1}, NEWLINE\]NEWLINENEWLINENEWLINEwhere \(f_j\) and \(l_j\) are respectively the \(j\)th Fibonacci and Lucas sequence terms. It follows from the definitions that the matrices \(F\) and \(L\) are Toeplitz matrices.NEWLINENEWLINEThe main results of this paper express the spectral norm of the matrices \(F\) and \(L\) in terms of Fibonacci and Lucas numbers, and state for \(n\) even thatNEWLINENEWLINENEWLINE\[NEWLINE \|F\|=\frac{1}{\sqrt{2}}\|F\|_2=\sqrt{\sum_{i=0}^{n-1}f_i\,f_{i+1}}=f_n,\qquad \|L\|=\sqrt{5}\|F\|=\sqrt{5}f_n, NEWLINE\]NEWLINENEWLINENEWLINEand for \(n=2t+1\) odd thatNEWLINENEWLINENEWLINE\[NEWLINE \|F\|=\frac{1}{\sqrt{2}}\|F\|_2=\sqrt{\sum_{i=0}^{n-1}f_i\,f_{i+1}}=\sqrt{f_n^2-1}, NEWLINE\]NEWLINENEWLINENEWLINEwith \(\|L\|=l_t l_{t+1}\) for even \(t\), and \(\|L\|= 5f_t f_{t+1}\) for odd \(t\).NEWLINENEWLINEA corollary for the Lucas matrices extends these results to all Schatten-\(p\) norms with \(0\leq p<\infty\), and \(n=2t+1\), such thatNEWLINENEWLINENEWLINE\[NEWLINE \|L\|_p=\left (l_t^p l_{t+1}^p+5^p f_t^p f_{t+1}^p\right )^{1/p}. NEWLINE\]NEWLINENEWLINENEWLINEAs a by-product of these results, it is demonstrated that the Fibonacci and Lucas matrices also obey analogous identities to those for the Fibonacci and Lucas numbers. In particular, it is shown thatNEWLINENEWLINENEWLINE\[NEWLINE \left (F^*\, F\right )^2=f_n\left (F^*\,F\right ),\qquad \text{and}\qquad L^*\, L=5F^*\, F, NEWLINE\]NEWLINENEWLINENEWLINEand the algebra constructed from such Toeplitz matrices is discussed.NEWLINENEWLINEConcluding results concern the spectral norms of circulant matrices constructed from the Fibonacci and Lucas sequences.