Constructing private matrix for Algebraic Eraser Having look into Algebraic Eraser key exchange (AEDH) seems the private key matrix may not be random to prevent weak-key results from the key exchange.

From the documentation and articles available it is not really clear for me how to construct the private key matrix. This session paper states:

$M_A = \sum_{i=0}^{N-1} \alpha_i . M_i^0$

however I am unable to find what is meant by $\alpha_i$ and $M_i^0$

For my spare-time projects maybe I will be ok with a random private key matrix $M_A$, but when I'm learning something, lets do it properly.

Thank you in advance.
You copied the equation wrong! I just checked your link.

The matrix $M_0$ is chosen from the set of $n\times n$ invertible matrices over $GF(q)$ (a group under multiplicatin) by some "proprietary" method.

A private key polynomial 
$$\sum_{i=0}^{N-1} \alpha_i x^i,$$
over $GF(q)$ is specified and $M_A$ is this polynomial evaluated at the matrix $M_0:$

$$M_A = \sum_{i=0}^{N-1} \alpha_i  M_0^i$$