What is the difference between the anti-image covariance and the anti-image correlation? What is the difference between the anti-image covariance and the anti-image correlation? How are the matrices of these coefficients computed, and what is the meaning of their elements?
 A: A excerpt of another answer (about factor analysis), structured.
Let $\bf R$ be a correlation or covariance matrix, and $\bf D$ be the diagonal matrix comprised of the inverses of diagonal elements of $\bf R^{-1}$. Then


*

*$\bf DR^{-1}D$ is known as anti-image covariance matrix of $\bf R$.
Its off-diagonal entries are the negatives of the partial covariance
coefficients (between two variables controlled for all the other
variables). The diagonal is equal to the diagonal $\bf D$, - these
diagonal values are called anti-images in $\bf R$.

*$\bf (DR^{-1}D)-2D+R$ is called image covariance matrix of $\bf R$.
Its diagonal entries are called images in $\bf R$ (they are equal
to the diagonal of $\bf R-D)$. An image is $R_i^2 \sigma_i^2$,
where $R_i^2$ is the squared multiple correlation coefficient of
dependency of variable $i$ on the rest variables, and $\sigma_i^2$ is
the diagonal element in $\bf R$, the variance (or $1$, in case of
correlation matrix).

*From the above it becomes clear that image + anti-image = $\sigma_i^2$, and that the two are the portions of a variable's variation being, respectively, explained and unexpained (residual) by the other variables. Thus, if $\bf R$ is correlations then image is the $R_i^2$ and anti-image is $1-R_i^2$; while if $\bf R$ is covariances then image is $R_i^2 \sigma_i^2$ and anti-image is $\sigma_i^2-R_i^2\sigma_i^2=\sigma_i^2(1-R_i^2)$.

*Terminologic warning: the image and anti-image covariance matrices
bear label "covariance" irrespective of whether $\bf R$ is
covariances or correlations.

*Anti-image correlation matrix of $\bf R$ is computed from anti-image covariances the very usual way like we convert usual covariance
into usual correlation, $r_{ij}=cov_{ij}/(\sigma_i \sigma_j)$, - i.e. here the cov and the two sigmas will be the values from an anti-image covariance matrix. Or in matrix notation: $\bf D^{-1/2} A D^{-1/2}$, where $\bf A$ is the anti-image covariance matrix and $\bf D^{-1/2}$ is its diagonal, square-rooted and inversed. Equivalent formula also is $\bf D^{1/2} R^{-1} D^{1/2}$, where $\bf D^{1/2}$ is the $\bf R^{-1}$'s diagonal, square-rooted and inversed. Off-diagonal
elements of anti-image correlation matrix are the negatives of the partial correlation
coefficients (between two variables controlled for all the other
variables). And that is popular way to compute partial correlations.

*One can also convert, analogously, image covariance matrix into image
correlation matrix, if needed.

*Anti-image correlation matrix will be the same - be $\bf R$
covariances or correlations (while anti-image covariance matrix was
different in the two instances).
