Motivated by Federico's recent question: Norm of upper triangular matrix of all ones I decided to ask a question about the norm of a matrix that is, in a sense, dual to the all ones upper triangular matrix.
Let $C_n$ be the $n \times n$ matrix (transpose of the Cesàro matrix)
$$ [C_n]_{ij} = \begin{cases} 1/j & i \le j\\ 0 & i > j \end{cases} $$
What is $\|C_n\|_2$?
Notes:
- From results on Cesàro operators, it seems to follow that $\|C_n\| \le 2$ for all $n \ge 1$.
- It is also worth noting that
$\|C_n\|_1=1$and$\|C_n\|_{\infty} = H_n$, where $H_n$ denotes the $n$-th Harmonic number. - Computing $\frac{e^TC_n^TC_ne}{e^Te}=2-H_n/n$ (e is the all ones vector) we obtain a simple lower-bound on $\|C_n\|_2$.
