# SVD of a matrix normal: practical applications?

What are some practical applications of the distributions of the components of an SVD of a matrix of normals?

In particular, assume $Y \sim N_{n \times p}({\bf 0}, \Sigma \otimes I)$, i.e. the rows of $Y$ are iid $N({\bf 0},\Sigma)$.

Then the SVD of ${\bf Y = UDV}^T$. We can study the marginal distributions of ${\bf U}$, ${\bf D}$, and ${\bf V}$. For example, I think ${\bf V}$ is known as the generalized Bingham distribution. It has some properties, such as $\mathbf{E}[{\bf V}] = {\bf 0}$ (the proof of which I cannot figure out so any references would be appreciated).

My question is: what are some reasons/applications of studying these distributions?

• Sampling properties of PCA estimators? – GeoMatt22 Feb 16 '17 at 2:18
• Right. So knowing the marginal distribution of U, say, allows me to make statements about the population given some sample. But its still unclear to me how to actually use the information, say, found in the jth singular vector of U. If it's right skewed or left skewed, what does that say about the population? – ilanman Feb 16 '17 at 12:20