Let $M\_n$ be the $n\times n$ matrix
$$
(M\_n)\_{ij}=\begin{cases}1 & i\leq j,\\\\ 0 &i>j.\end{cases}
$$
Is there around an explicit expression or at least an asymptotic for $\left\Vert M\_n \right\Vert$? The norm is the usual Euclidean induced norm $\left\Vert M \right\Vert=\rho(M^TM)^{1/2}$.


I apologize if this a stupid question...



2-norm of the upper triangular "all-ones" matrix
The eigenvalues of $M^{\rm T}M$ are $1 / (4 \phantom. \cos^2\frac{k\pi}{2n+1})$ for $k=1,2,\ldots,n$. The largest of these arises for $k=n$ and equals $1/(4\phantom.\sin^2\frac{\pi}{4n+2})$. Hence $\|M\| = 1 / (2 \phantom.\sin\frac{\pi}{4n+2})$, which is asymptotic to $2n/\pi$. This is easier to see if we work not with $M$ but with its inverse, which is a unipotent matrix with $-1$'s on the first subdiagonal and $0$'s elsewhere.


**EDIT** Dividing by $n$ and letting $n \rightarrow \infty$, we also recover a form of 
Wirtinger's inequality: the operator $T$ on $L^2(0,1)$ taking a function $f$ to its indefinite integral [i.e. $Tf(x) = \int\_0^x f(y) \phantom. dy$] has norm $2/\pi$, attained by $f(x) = \cos (\pi x /2)$. [To see the connection, compare the Riemann sums for $\|f\|\_2^2 = \int\_0^1 f(x)^2 dx$ and $\|Tf\|\_2^2 = \int\_0^1 (\int\_0^x f(y) \phantom. dy)^2 dx$.]


