# Reducing to a matrix

## Main Question or Discussion Point

I'm trying to find a way of reducing a rank-3 tensor field into a matrix, but am having trouble finding a good way to do it. The situation set up is as follows:

Let's say that I have a $3\times 3$ matrix and a $5 \times 1$ vector as follows

$$A(x) = \begin{pmatrix} a_{11}(x) & a_{12}(x) & a_{13}(x) \\ a_{21}(x) & a_{22}(x) & a_{23}(x) \\ a_{31}(x) & a_{32}(x) & a_{33}(x) \end{pmatrix}, \qquad \qquad x = \begin{pmatrix} x_1 \\ x_2\\ x_3 \\ x_4 \\ x_5 \end{pmatrix}$$

Now I want to find a tangent plane to a surface defined by A(x) at some point $\bar x$ to create some constraints, namely, I want to do something of the form

$$\nabla A(x) (x - \bar x) = 0$$

Now the problem here is that $\nabla A(x)$ is a rank 3 tensor. I need to find some what of ''matricizing'' this tensor into a matrix so that I can solve a linear programming problem with it. Any ideas?