# Is this Matrix Normal Distribution correctly defined?

In this paper, the following appears

It's stated that $$V^{-1}$$ is the row covariance matrix for the Matrix-Normal.

However, according to this link in wikipedia, and the references therein, and adapting their notation to the above, $$V^{-1}$$ should be the column covariance matrix instead.

Am I correct, or have I made a mistake? If so, I would like some help to understand where.

The second convention is to string out the columns. This is done with the $$\text{vec}$$ operator, which is more common in other areas of mathematics.
Obviously these ideas aren't totally separate. For example, if you want to use formulas with say Kronecker products that apply when you're using the traditional $$\text{vec}$$ operator, just take the transpose of your data matrix beforehand.