# A Tutorial on the Spectral Theory of Markov Chains

arXiv: https://arxiv.org/abs/2207.02296 · HTML: https://mlsanity.com/paper/2207.02296/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 106 references that are in the map.
- First posted on arXiv 5 Jul 2022.

## Builds on (most cited first, 5 of 5)

- [The Laplacian in RL: Learning Representations with Efficient Approximations](https://mlsanity.com/paper/1810.04586/) (2018)
- [Representations for Stable Off-Policy Reinforcement Learning](https://mlsanity.com/paper/2007.05520/) (2020)
- [Signal Processing on Directed Graphs](https://mlsanity.com/paper/2008.00586/) (2020)
- [Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey](https://mlsanity.com/paper/2106.02154/) (2021)
- [Laplacian Matrix for Dimensionality Reduction and Clustering](https://mlsanity.com/paper/1909.08381/) (2019)

## Built on by (most cited first, 2 of 2)

- [Modes of Sequence Models and Learning Coefficients](https://mlsanity.com/paper/2504.18048/) (2025)
- [Parametrized Power-Iteration Clustering for Directed Graphs](https://mlsanity.com/paper/2210.00310/) (2022)

## Abstract (arXiv)

Markov chains are a class of probabilistic models that have achieved widespread application in the quantitative sciences. This is in part due to their versatility, but is compounded by the ease with which they can be probed analytically. This tutorial provides an in-depth introduction to Markov chains, and explores their connection to graphs and random walks. We utilize tools from linear algebra and graph theory to describe the transition matrices of different types of Markov chains, with a particular focus on exploring properties of the eigenvalues and eigenvectors corresponding to these matrices. The results presented are relevant to a number of methods in machine learning and data mining, which we describe at various stages. Rather than being a novel academic study in its own right, this text presents a collection of known results, together with some new concepts. Moreover, the tutorial focuses on offering intuition to readers rather than formal understanding, and only assumes basic exposure to concepts from linear algebra and probability theory. It is therefore accessible to students and researchers from a wide variety of disciplines.