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I need to apply a differential operator (nabla) on a matrix. Problem is, that I don't know how to calculate that. Do I treat nabla as a column vector and simply multiply vector with the matrix? Or is nabla to be handled as jacobi matrix?

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I've never seen the standard del operator $\nabla$ applied to a matrix. What is the context of the application? – Emily Aug 23 '12 at 18:05
There isn't really any difference between the $\nabla$ and Jacobi-matrix (up to transposition), it only depends on where your function, defined on matrices, maps to. – Cocopuffs Aug 23 '12 at 18:11
It depends on what you mean by applying a differential operator to a matrix. What kind of entries does the matrix have? What do you want this differential operator to do to it? – Qiaochu Yuan Aug 23 '12 at 18:12
Context is the nonlinear sigma model in quantum field theory, where I want to use a matrix function $\phi$ instead of a vector function. The matrix is a 3x3 one with simple scalar functions in it. – Tornado Aug 23 '12 at 18:13
Treat it the same as a vector function, just with the entries in a different arrangement. – Cocopuffs Aug 23 '12 at 18:17

The differential operators can be defined in a matrix of differential operators, so that, for instance, it could be applied to a matrix containing n vectors, one in each column.

$$ \left[\begin{matrix} \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}}\\ \end{matrix}\right] \left[\begin{matrix} x_1 & x_2 & x_3 & & x_n \\ y_1 & y_2 & y_3 & \dots & y_n \\ z_1 & z_2 & z_3 & & z_n \\ \end{matrix}\right] $$

To see how to implement this kind of operator in the Python mudule Sympy, you can refer to here. To see more discussion about matrix of differential operator, see here.

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