Is it possible to compute the norm $$ ||A||=\sup_{X\in \mathbb{R}^n\setminus \{0\}} \frac{\lVert AX\rVert}{\lVert X\rVert} $$ where $A$ is the matrice defined by $AX=(x_1-x_n,x_2-x_1,\ldots,x_n-x_{n-1})$?
I can compute it given the norm is defined as the Frobenius norm: $\lVert X\rVert=^t X X$ and we obtain the square root of the maximum of the eigenvalues of $^t A A$. But if we don't specify the norm $\lVert\cdot \rVert$ is it possible to determine $\lVert \cdot \rVert$ ?