Linear algebra fact

Here is interesting linear algebra fact: let be an matrix and be a vector such that . Then for any matrix , .

The proof is just basic algebra: .

Why care about this? Let’s imagine that is a (not necessarily symmetric) stochastic matrix, so . Let be a low-rank approximation to (so consists of all the large singular values, and consists of all the small singular values). Unfortunately since is not symmetric, this low-rank approximation doesn’t preserve the eigenvalues of and so we need not have . The can be thought of as a “correction” term such that the resulting matrix is still low-rank, but we’ve preserved one of the eigenvectors of .