On the norms of circulant matrices with the \((k,h)\)-Fibonacci and \((k,h)\)-Lucas numbers (Q2892557)

For any \(k>0\) and \(h<-1\) the \((k,h)\)-Fibonacci numbers \(F_{n}^{(k,h)}\) and the \((k,h)\)-Lucas numbers \(L_{n}^{(k,h)}\) are given by \((\alpha^{n}-\beta ^{n})/(\alpha-\beta)\) and \(\alpha^{n}+\beta^{n}\), respectively, where \(\alpha>\beta\) are the (real) roots of the equation \(x^{2}-kx+h=0\). Consider the \(n\times n\) circulant matrices \(A_{n}\) and \(B_{n}\) whose first rows are \([F_{0}^{(k,h)},F_{1}^{(k,h)},...,F_{n-1}^{(k,h)}]\) and \([L_{0}^{(k,h)} ,L_{1}^{(k,h)},...,L_{n-1}^{(k,h)}]\), respectively. The authors compute upper and lower bounds for the induced \(2\)-norms \(\left\| A_{n}\right\| _{2}\) and \(\left\| B_{n}\right\| _{2}\).NEWLINENEWLINEReviewer's remark: The \(2\)-norms can be computed exactly as follows. The \(2\)-norm of a normal matrix is equal to the largest of the absolute values of its eigenvalues. Circulants are normal matrices and the conditions on \(k\) and \(h\) ensure that the entries of the matrices \(A_{n}\) and \(B_{n}\) are nonnegative. Since \([1,1,...,1]^{T}\) is an eigenvector for \(A_{n}\), Frobenius' theorem for nonnegative matrices shows that \(\left\| A_{n}\right\| _{2}\) is equal to the corresponding eigenvalue which is equal to each of the row-sums of \(A_{n}\) (and similarly for \(B_{n}\)).