 EM update for a single-factor Gaussian model

In factor analysis with one factor and fixed noise variance $v>0$, let $\lambda$ be the current loading vector, $s=v+|\lambda {|}^2$, and $S=n^{-1}{\sum }_i{Y}_i{Y}_i^T$. The expectation-maximization algorithm update isThe conditional factor means are ${\lambda }^T{Y}_i/s$ and their variances are $v/s$; maximizing the expected complete-data log-likelihood gives the formula.