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A number of areas I'm studying in my degree (not a maths degree) involve eigenvalues and eigvenvectors, which have never been properly explained to me. I find it very difficult to understand the explanations given in textbooks and lectures. Does anyone know of a good, fairly simple but mathematical explanation of eigenvectors and eigenvalues on the internet? If not, could someone provide one here?

As well as some of the mathematical explanations, I'm also very interested in 'big picture' answers as to why on earth I should care about eigenvectors/eigenvalues, and what they actually 'mean'.

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The reasons you might care could depend on what areas you're studying that involve eigenvalues and eigenvectors. Could you please share what those areas are, and possibly what the involvement is? – Jonas Meyer May 3 '11 at 23:27
Given your programming background, I imagine PageRank would be a little interesting for you: $$ $$ – Will Jagy May 3 '11 at 23:32
I would recommend looking at… -- the topic is pretty well covered in answers as well as comments, so hopefully it will be helpful. – InterestedGuest May 3 '11 at 23:32
I came across this "tutorial" which covers basic matrix computations, etc., and ends with discussion/explanation of eigenvalues and eigenvectors. Scroll down the page for those links...<>; – amWhy May 4 '11 at 0:18
up vote 43 down vote accepted

To understand why you encounter eigenvalues/eigenvectors everywhere, you must first understand why you encounter matrices and vectors everywhere.

In a vast number of situations, the objects you study and the stuff you can do with them relate to vectors and linear transformations, which are represented as matrices.

So, in many many interesting situations, important relations are expressed as $$\vec{y} = M \vec{x}$$ where $\vec{y}$ and $\vec{x}$ are vectors and $M$ is a matrix. This ranges from systems of linear equations you have to solve (which occurs virtually everywhere in science and engineering) to more sophisticated engineering problems (finite element simulations). It also is the foundation for (a lot of) quantum mechanics. It is further used to describe the typical geometric transformations you can do with vector graphics and 3D graphics in computer games.

Now, it is generally not straight forward to look at some matrix $M$ and immediately tell what it is going to do when you multiply it with some vector $\vec{x}$. Also, in the study of iterative algorithms you need to know something about higher powers of the matrix $M$, i.e. $M^k = M \cdot M \cdot ... M$, $k$ times. This is a bit awkward and costly to compute in a naive fashion.

For a lot of matrices, you can find special vectors with a very simple relationship between the vector $\vec{x}$ itself, and the vector $\vec{y} = Mx$. For example, if you look at the matrix $\left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$, you see that the vector $\left(\begin{array}{c} 1\\ 1\end{array}\right)$ when multiplied with the matrix will just give you that vector again!

For such a vector, it is very easy to see what $M\vec{x}$ looks like, and even what $M^k \vec{x}$ looks like, since, obviously, repeated application won't change it.

This observation is generalized by the concept of eigenvectors. An eigenvector of a matrix $M$ is any vector $\vec{x}$ that only gets scaled (i.e. just multiplied by a number) when multiplied with $M$. Formally, $$M\vec{x} = \lambda \vec{x}$$ for some number $\lambda$ (real or complex depending on the matrices you are looking at).

So, if your matrix $M$ describes a system of some sort, the eigenvectors are those vectors that, when they go through the system, are changed in a very easy way. If $M$, for example, describes geometric operations, then $M$ could, in principle, stretch and rotate your vectors. But eigenvectors only get stretched, not rotated.

The next important concept is that of an eigenbasis. By choosing a different basis for your vector space, you can alter the appearance of the matrix $M$ in that basis. Simply speaking, the $i$-th column of $M$ tells you what the $i$-th basis vector multiplied with $M$ would look like. If all your basis vectors are also eigenvectors, then it is not hard to see that the matrix $M$ is diagonal. Diagonal matrices are a welcome sight, because they are really easy to deal with: Matrix-vector and Matrix-matrix multiplication becomes very efficient, and computing the $k$-th power of a diagonal matrix is also trivial.

I think for a "broad" introduction this might suffice?

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This is a question I've had myself for many years, so thanks for the excellent summary. Would you have any references that expands the general reasoning you have put forward here? – daven11 May 24 '11 at 13:06
I've found lately that eigenvalue 1 is especially interesting because if it is present some fixed points for transformation appear as in this example above. – Widawensen Jun 17 at 11:03

This made it clearer for me: Khan Academy - Introduction to Eigenvalues and Eigenvectors I often find it easier to understand via illustration like this.

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link is now busted – tim peterson Sep 17 '14 at 23:19
Updated it, but not sure if they like external links as answers. – Amala Sep 18 '14 at 15:03
Including a link is good, but you should also summarize the content of the linked document if possible. – Nate C-K Apr 22 '15 at 10:54

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