# Self outer product of a vector that is itself a sum of vectors

Suppose there are three vectors a, b, and c. The data that I observe in my experiment is vector $v=a+b+c$ (i.e. I can not observe $a,b,c$ independently). Now suppose I take the outer product of vector $v$ giving me the matrix $K=vv^T=(a+b+c)(a+b+c)^T$. Is there a way for me to remove the cross-terms within K (e.g. remove $ab^T$, etc.) such that I am left with only the self outer-product terms? Ultimately, I would like this to work for the case where $v$ is the sum of an arbitrary number of vectors.

• Why don't you just subtract off the cross terms? Commented Jun 17, 2015 at 16:28
• Or, do you mean that you want to find a vector $w$ such that $$ww^T = aa^T + bb^T + cc^T?$$ Commented Jun 17, 2015 at 16:28
• Yes, that is exactly what I would like to find! But I do not know if there is a way to do this considering that I do not have individual measurements of a,b,c. I only know the sum v=a+b+c. Commented Jun 17, 2015 at 16:30
• You will (in general) need at least $3$ measurements of some kind if you want to build the matrix using outer products. Commented Jun 17, 2015 at 16:36

Note that every outer product is a rank-1 matrix (and conversely, every symmetric rank-1 matrix is an outer product). The matrix $M = aa^T + bb^T + cc^T$ could have a rank of $3$.
If $M$ is symmetric with a rank of $r$, then we could only choose vectors $u_1,\dots,u_k$ such that $M = u_1u_1^T + \cdots + u_k u_k^T$ if $k \geq r$.
$$(a +b + c)(a + b +c)^T = a(a+b+c)^T + b(a+b+c)^T + c (a+b+c)^T = aa^T + bb^T + cc^T + ab^T + ac^T + ba^T + bc^T + ca^T + cb^T.$$