Principal component analysis basically takes the longest eingenvectors $\vec u_i$ of covariacne matrix $C_{ij}= \sum_k X_{ki} X_{kj} $ where $X_{ki}$ is $i$-th component of $k$-th data sample from some dataset $X$.
I'm interested if there is something similar possible to do for mutual covariance matrix $M_{ij}= \sum_k X_{ki} Y_{kj} $ where $X$ and $Y$ are different datasets (of the same dimension, if it makes situation easier).
While principal components of $C_{ij}$ says how big is change $x_j$ on change of $x_i$ (i.e. correlation between two components of data from dataset $X$), it would say how big is change of $y_j$ on change of $x_i$ (i.e. correlation between component of data samples from dataset $X$ and an other component of data samples from dataset $Y$ ).