# How does “Fundamental Theorem of Factor Analysis” apply to PCA, or how are PCA loadings defined?

I’m currently going through a slide set I have for “factor analysis” (PCA as far as I can tell).

In it, the “fundamental theorem of factor analysis” is derived which claims that the correlation matrix of the data going into the analysis ($\bf R$) can be recovered using the matrix of factor loadings ($\bf A$):

This however confuses me. In PCA the matrix of “factor loadings” is given by the matrix of eigenvectors of the covariance/correlation matrix of the data (since we’re assuming that the data have been standardized, they are the same), with each eigenvector scaled to have length one. This matrix is orthogonal, thus $\bf AA^\top = I$ which is in general not equal to $\bf R$.

Indeed, if the eigen-decomposition of the correlation matrix is where $\mathbf V$ are eigenvectors (principal axes) and $\mathbf S$ is a diagonal matrix of eigenvalues, and if we define loadings as then one can easily see that Moreover, the best rank-$r$ approximation to the correlation matrix is given by the first $r$ PCA loadings: