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## gradient of row vector multiplied by scalar

I'm trying to re-write $v (u x)$ where $v$ and $u$ are row vectors and $x$ is a column vector as some expression $M x$ (or $\bar{v}x$, etc.). The motivation is because I'm trying to compute the gradient (and $\nabla^{'}_x [ v (u x) ] \cdot x$ should yield a row vector).

Intuitively, since the expression $v (ux)$ can be re-written as

$[v_1 (ux), v_2(ux), \cdots, v_n(ux)]$

(where $v_i$ is a scalar), this quantity can be computed. However, doing this and substituting into a larger expression is pretty obnoxious and I'd like to do it using standard matrix operations (non element-wise) if possible.

Any help is appreciated.

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I don't think this is appropriate for MO. That being said, the first thing to note is that you should be consistent that vectors are always columns or always rows. Let's make the convention that vectors are always written as rows. Then, the main point is that in matrix notation the inner product $x \cdot y$ is simply $x y^T$. – Tony Huynh Jan 12 2011 at 22:22
math.stackexchange.com – David Roberts Jan 12 2011 at 23:13
$A=\nabla_{x}[v(ux)]$ is a four-component tensor with elements $A_{ijkl}=\nabla_{x_{i}}v_{j}u_{k}x_{l}=\delta_{il}v_{j}u_{k}$; is there anything more to say? – Carlo Beenakker Jan 13 2011 at 8:47