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In this paper, the following appears

enter image description here

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.

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There are two conventions used when defining normal random matrices. They both, however, make use of the definition of normal random vectors.

The first convention is to string out the rows of a matrix. I recall this was what was done in a multivariate statistics class I took a few years ago. This makes sense because data are often found in this format. Each row is independent of every other row. And the columns in each row are dependent in the same way. For example, see the bottom of page 52 in this book.

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.

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