Notation for a matrix times nabla

Question

Let $A$ be a matrix of $\mathbb{R}^{n\times n}$ and $u : \mathbb{R}^n \to \mathbb{R}^n$ be a vector function. What does the notation

$$(A\nabla)u$$

mean?

Answer 1

Usually it would be something like that , and i hope this is helpful.

TO give an example in $\mathbb{R}^{2 \times2}$
$$ \begin{pmatrix} a && b\\c && d \end{pmatrix} \begin{pmatrix} \partial_{1} \\ \partial_{2} \end{pmatrix} = \begin{pmatrix} a\partial_{1}+b \partial_{2} \\ c \partial_{1} + d \partial_{2} \end{pmatrix}$$ and times u : $$ \begin{pmatrix} a\partial_{1}+b \partial_{2} \\ c \partial_{1} + d \partial_{2} \end{pmatrix} \begin{pmatrix} u_{1} \\ u_{2} \end{pmatrix} = (a\partial_{1}+b \partial_{2})u_{1} + (c \partial_{1} + d \partial_{2})u_{2} $$ But $u_{1}$ and $u_{2}$ need to be functions, or it gives zero...

If this wasn't helpful, may you give us some context ? Like the entire question ?

Answer 2

The "standard" way of interpreting this is that $\nabla u$, when $u$ is a vector, generates a tensor of the following form (using index notation): $$\nabla u = \frac{\partial u_i}{\partial x_j}$$

Consequently, $A$ is a linear operator/matrix acting on $\nabla u$, which is well-defined under the rules of matrix multiplication since they both have the same "dimensionality". Again using index notation, this multiplication looks like:

$$\left[(A\nabla)u\right]_{kj} = A_{ki}\frac{\partial u_i}{\partial x_j}$$