# Interpreting Eigenvalues of Co-variance Matrix

Im working on market reaction to events and I'm using the co-variance matrix to do this. In this paper the author writes

It has been known for some time that the largest eigenvalue (λ1) contains information on the risk associated with the particular assets of which the co-variance matrix is comprised.

However, there's no reference for this and I haven't found anything that backs up his point because eigenvalues are used for many other things.

What I want to know is:

1. How do we know that the first eigenvalue contains the most information?
2. Where can I get more to read on this topic?
• Here is a quote from "what is the application of eigenvalues in statistics?": "If you calculate the eigenvectors of a covariance matrix the first eigenvector is the axis of greatest variance in the data (and this variance is given by the first eiegnvalue). The next eigenvector is the axis of next greatest variance, and so on." This is a basic result in multivariate statistics. – noob2 Jul 29 '16 at 12:54

What you basically do here is a Principal Component Analysis (PCA). A good starting point in the financial sphere is

Page 3:
"The most natural choice of uncorrelated risk sources is provided by the principal component decomposition of the returns covariance [...] The eigenvectors define a set of N uncorrelated portfolios, the principal portfolios [...] are decreasingly responsible for the randomness in the market. Indeed, the eigenvalues correspond to the variances of these uncorrelated portfolios."

On page 4, Figure 1 comes a geometric interpretation which should make things intuitive: NB: Risk is also defined as volatility here! (as often in finance)

• @AdegokeA: Thank you, a warm welcome to Quant.SE and thank you for your interesting question :-) – vonjd Jul 29 '16 at 13:12
• @vonjd By the way, could you suggest me a good book where PCA is treated? I would like a rigorous mathematical treatment. Apparently I haven't found any of the sort yet. Thanks. – RandomGuy Jul 29 '16 at 14:21
• @RandomGuy: I would recommend Attilio Meucci's book: symmys.com/attilio-meucci/book and for PCA therein here: symmys.com/node/196 (p. 138ff.) – vonjd Aug 1 '16 at 13:11
• @vonjd Thank you very much. I am looking for a good mathematical introduction to PCA and so far I found only insufficient or mathematically light introductions to this topic. – RandomGuy Aug 1 '16 at 13:36