We know that principle component analysis (PCA) is a eigenvalue problem. Let $A$ be the covariance matrix of $X$, PCA aims to find the eigenvalue of $A$:$\max v'Av$, subject to $v'v=1$ Multiple PCs can be found by deflation:Let $\hat{X}=X-\sum_{i=1}^k v_iv_i'X$ (or equivalently $\hat{A}=(I-\sum_{i=1}^k v_iv_i')A(I-\sum_{i=1}^k v_iv_i')$) where $v_i, i = 1,\dots,k$ is the first $k$ PCs and then the $k+1$st PC can be computed by solving the $1$st PC of $\hat{A}$.I am wondering if there is a similar deflation method to find the first $k$ eigenvectors for the generalized eigenvalue problem:$\max \frac{v'Av}{v'Bv}$where $A$ and $B$ are some covariance matrices: $A=Cov(X), B=Cov(Y)$.Thanks!
Deflation for generalized eigenvalue problem
optimization;eigenvalues