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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.

In factor analysis, the independent multivariate normal distributions of $Z_i$ and ${\xi }_i$ imply that their linear combination is also multivariate normal. Its expected value is zero and its covariance matrix isThereforeThe noise variance is assumed positive. The factor contribution is positive semidefinite and has rank at most $k$, while the noise makes the marginal covariance nonsingular.