Linear Algebra – Is the Norm of the Hilbert Matrix Equal to $\pi$?

Question

Let $A$ be a Hilbert matrix,

$$a_{ij}=\frac{1}{1+i+j}$$

We have the result $\| A \| \leq \pi$. I am using the subordinate norm of the Euclidean norm, i.e.,

$$\| A \| = \sup\{\langle Ax,y\rangle:\quad x,y\in\mathbb{R}^n,\quad\Vert x\Vert_2\leq 1,\quad\Vert y\Vert_2\leq 1\}$$

This inequality can be proved using Hilbert's Inequality. Look here.

Question: Do we have an equality? I found nothing on the Internet about such equality.

Answer 1

Certainly not (if you're talking about a finite matrix, rather than an operator on $\ell^2$). $\pi$ is transcendental, and the norm of a finite matrix with rational entries is an algebraic number.