# Understanding equality regrading expectation of random matrices

I'm reading the following article on Latent Tree Structures (I added a link at the end of the post) : "Spectral Methods for Learning Multivariate Latent Tree Structure".

I'm trying to understand the following equality (w,v,u are random vectors) : https://i.stack.imgur.com/RR7BS.png

I don't get both equalities. the following assumption holds -

1. The following equality holds between w and v : E[v|w] = A(v|w)*w for a fixed matrix A(v|w).

But even with this assumption I can't see how can you get the previous mentioned equality.

Thanks for the help!

The article : https://arxiv.org/pdf/1107.1283.pdf