### Category: Linear Algebra

### Davis-Kahan Sin Theta Theorem

In many situations, there is a symmetric matrix of interest , but one only has a perturbed version of it . How is affected by ?The basic example is Principal Component Analysis (PCA). Let for some random vector , and let be the sample covariance matrix on independent copies of . Namely, we observe i.i.d.

### Perron Frobenius Theorem

Perron Frobenius Theorem is a well-known algebra result that finds applications in a large span of fields of Mathematics. One can mention Markov chains, population growth (with the Leslie matrix model) or the famous PageRank algorithm. In this post we prove the Perron-Frobenius Theorem for stochastic matrices. Perron Frobenius Theorem Let us consider some integer