I have the following matrix,
$$ \mathbf{A} \in \mathbb{R}^{n\times n}: \ a_{i, j} = \min\{i, j\} - \frac{i j}{n+1} $$
and I was wondering if it was possible to rewrite it in terms of the vector,
$$ \mathbf{[n]} = \begin{pmatrix} 1 \\ 2 \\ 3 \\ \vdots \\ n \end{pmatrix} $$.
So far...
... I notice that
$$ \mathbf{A} = \begin{pmatrix} 1 & 1 & 1 &\ldots \\ 1 & 2 & 2 &\ldots \\ 1 & 2 & 3 &\ldots\\ \vdots & & & \ddots \end{pmatrix} - \frac{1}{n+1} \mathbf{[n]} \mathbf{[n]}^{\prime} $$
but, after looking into its LU decomposition, I could not work out if that first matrix $\min\{i, j\}$ can be written in terms of $\mathbf{[n]}$.