Topic: Jordan Normal Form, Spectral Radius and Positive Matrices

Date: Nov. 22, Nov 29 and Dec 6

Presenter: Liran Li

Source Material: https://iuuk.mff.cuni.cz/~rakdver/linalg/lesson15-9.pdf, which is lecture notes from a linear algebra course, and https://www.math.upenn.edu/~moose/240S2013/slides7-31.pdf, which serves as an introduction to a particular definition of generalized eigenvectors.

Any square complex matrix is similar to its Jordan normal form. Starting from its Jordan normal form, we are able to prove a lot of things about an element-wise non-negative matrix that becomes element-wise positive when raised to some power. This series of presentations are an adaptation of the lecture notes by Prof. Zdeněk Dvořák at Computer Science Institute of Charles University, Prague. We discuss some less familiar results about these positive marices, which then lead to a potential use case — PageRank.