Outer product is defined as $\langle a,b \rangle = a b^T$ where $a, b \in \mathbb{R}^{n\times1}$. We have an $n \times n $ matrix $A$ whose entries are given by $a_{j,j+1}=1$ and all other elements are 0. For example, $ A = \left[\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix} \right] $
How do I write such a matrix as sum of outer-products only (no quadratic forms, etc.) ? I think it involves the canonical vectors $e_j = [0,\ldots,0,1,0, \ldots, 0]$ but can't see how. Any hints ?