eigen values and Page Rank

cs algebra

Learnt from: No One Taught Eigenvalues & EigenVectors Like This - YouTube

• The Page Rank algorithm calculates the link matrix (a.k.a the Google matrix) of website
  • this is a left stochastic matrix (Markov Matrix) so it always has an eigen value which is 1, per a comment in the video and wikipedia

Quote

A stationary probability vector $\pi$ is defined as a distribution, written as a row vector, that does not change under application of the transition matrix; that is, it is defined as a probability distribution on the set {1, …, n} which is also a row eigenvector of the probability matrix, associated with eigenvalue 1: $\pi P=\pi$

• The eigen vector corresponding to the eigen value 1 describes the ranking of the pages.
  • As this vector is unchanged during the application of the transition matrix, it can be intuitively thought as a probability distribution of popularity amongst all pages