Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
up vote 2 down vote accepted

$\langle e_i, e_j \rangle$ is a matrix with a $1$ in position $(i,j)$ and zeros elsewhere. So you can just add up three such terms, with $i$'s and $j$'s corresponding to the positions where you have $1$ in your matrix $A$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.