Principal component analysis (PCA) uses one scatter matrix (an estimation of covariance matrix) to find eigen vectors of the original matrix.
Suppose we have two scatter $V_1, V_2$ matrices for our observations (assume classical scatter and fourth momentum scatter matrices), some metrics for outliers like Invariant Coordinate selection (ICS) uses two scatter matrices to compute the invariant coordinates or components.
The goal of ICS is to find $p\times p$ matrix $B(X_n)$ and diagonal matrix $D(X_n)$, s.t.
$$B(X_n)V_1(X_n)B'\left(X_n \right) = I_p ~~~~~\& B(X_n)V_2(X_n)B'\left(X_n \right)=D(X_n)$$
$D(X_n)$ contains the eigenvalues of $V_1(X_n)^{-1}V_2(X_n)$ in decreasing order while $B(X_n)$ contains the corresponding eigen vectors.
Then using any affine equivariant location estimator $m(X_n)$, the corresponding scores are the invariant coordinates or components:
$$Z_n = (z_1,...,z_n)' = (X_n - 1_nm(X_n)')B(X_n)'$$
Question: In PCA, we use one covariance matrix (scatter matrix), what is the purpose of using two scatter matrices in ICS? How does using two scatter matrices affect getting the invariant coordinates? Is it because of the eigenvectors we get as a result of that, which are stored in $B(X_n)$?
