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What is the importance of eigenvalues/eigenvectors?

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en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors have you looked at this? It offers a pretty complete answer to the question. –  InterestedGuest Feb 23 '11 at 2:34
    
Here is a nice explanation: hubpages.com/hub/What-the-Heck-are-Eigenvalues-and-Eigenvectors –  PEV Feb 23 '11 at 2:41
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Huh. I am extremely surprised this question hasn't already come up. –  Qiaochu Yuan Feb 23 '11 at 3:06
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I realize this isn't my question, but I would love to see answers addressing the specific question, "How do you motivate eigenvalues and eigenvectors to a group of students who are only familiar with very basic matrix theory and who don't know anything about vector spaces or linear transformations?" –  Jason DeVito Feb 23 '11 at 4:52
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@Jason: Then you should post that as a question! –  Arturo Magidin Feb 23 '11 at 6:01
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6 Answers

up vote 71 down vote accepted

Short answer: Eigenvectors make understanding linear transformations easy. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenvalues give you the factors by which this compression occurs.

The more directions you have along which you understand the behavior of a linear transformation, the easier it is to understand the linear transformation; so you want to have as many linearly independent eigenvectors as possible associated to a single linear transformation.

Slightly longer answer: there are a lot of problems that can be modeled with linear transformations, and the eigenvectors give very simply solutions. For example, consider the system of linear differential equations \begin{align*} \frac{dx}{dt} &= ax + by\\ \frac{dy}{dt} &= cx + dy. \end{align*} This kind of system arises when you describe, for example, the growth of population of two species that affect one another. For example, you might have that species $x$ is a predator on species $y$; the more $x$ you have, the fewer $y$ will be around to reproduce; but the fewer $y$ that are around, the less food there is for $x$, so fewer $x$s will reproduce; but then fewer $x$s are around so that takes pressure off $y$, which increases; but then there is more food for $x$, so $x$ increases; and so on and so forth. It also arises when you have certain physical phenomena, such a particle on a moving fluid, where the velocity vector depends on the position along the fluid.

Solving this system directly is complicated. But suppose that you could do a change of variable so that instead of working with $x$ and $y$, you could work with $z$ and $w$ (which depend linearly on $x$; that is, $z=\alpha x+\beta y$ for some constants $\alpha$ and $\beta$, and $w=\gamma x + \delta y$, for some constants $\gamma$ and $\delta$) and the system transformed into something like \begin{align*} \frac{dz}{dt} &= \kappa z\\ \frac{dw}{dt} &= \lambda w \end{align*} that is, you can "decouple" the system, so that now you are dealing with two independent functions. Then solving this problem becomes rather easy: $z=Ae^{\kappa t}$, and $w=Be^{\lambda t}$. Then you can use the formulas for $z$ and $w$ to find expressions for $x$ and $y$.

Can this be done? Well, it amounts precisely to finding two linearly independent eigenvectors for the matrix $\left(\begin{array}{cc}a & b\\c & d\end{array}\right)$! $z$ and $w$ correspond to the eigenvectors, and $\kappa$ and $\lambda$ to the eigenvalues. By taking an expression that "mixes" $x$ and $y$, and "decoupling it" into one that acts independently on two different functions, the problem becomes a lot easier.

That is the essence of what one hopes to do with the eigenvectors and eigenvalues: "decouple" the ways in which the linear transformation acts into a number of independent actions along separate "directions", that can be dealt with independently. A lot of problems come down to figuring out these "lines of independent action", and understanding them can really help you figure out what the matrix/linear transformation is "really" doing.

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+1: This is the clearest explanation I have seen so far about the connection between systems of differential equations and eigenvectors - impressive! –  vonjd May 15 '11 at 10:23
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Sir. You are an amazing person. –  HowardRoark Jan 17 '13 at 19:08
    
Thanks for writing this. –  Surya Jul 24 '13 at 14:29
    
The example of predator is superb! Thanks. –  foresightyj Dec 2 '13 at 1:51
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A short explanation:

Consider a matrix $A$, for an example one representing a physical transformation. When this matrix is used to transform a given vector $x$ the result is $y = A x$.

Now an interesting question is are there any vectors $x$ which does not change it's direction under this transformation? However allow the vector magnitude to vary by scalar $ \lambda $.

Such a question is of the form $A x = \lambda x $.

So such special $x$ are called eigen vectors and the change in magnitude depends on the eigen value $ \lambda $.

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Good short answer!! –  gpuguy Jul 14 '12 at 9:25
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The behaviour of a linear transformation can be obscured by the choice of basis. For some transformations, this behaviour can be made clear by choosing a basis of eigenvectors: the linear transformation is then a (non-uniform in general) scaling along the directions of the eigenvectors. The eigenvalues are the scale factors.

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When you apply transformations to the systems/objects represented by matrices, and you need some characteristics of these matrices you have to calculate eigenvectors (eigenvalues).

"Having an eigenvalue is an accidental property of a real matrix (since it may fail to have an eigenvalue), but every complex matrix has an eigenvalue."(Wikipedia)

Eigenvalues ​​characterize important properties of linear transformations, such as whether a system of linear equations has a unique solution or not. In many applications eigenvalues ​​also describe physical properties of a mathematical model.

Some important applications -

  • Principal Components Analysis (PCA) in object/image recognition;

  • Physics - stability analysis, the physics of rotating bodies;

  • Market risk analysis - to define if a matrix is positive definite;

  • PageRank from Google.

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I think if you want a better answer, you need to tell us more precisely what you may have in mind: are you interested in theoretical aspects of eigenvalues; do you have a specific application in mind? Matrices by themselves are just arrays of numbers, which take meaning once you set up a context. Without the context, it seems difficult to give you a good answer. If you use matrices to describe adjacency relations, then eigenvalues/vectors may mean one thing; if you use them to represent linear maps something else, etc.

One possible application: In some cases, you may be able to diagonalize your matrix M using the eigenvalues, which gives you a nice expression for M^k. Specifically, you may be able to decompose your matrix into a product SDS^-1 , where D is diagonal, with entries the eigenvalues, and S is the matrix with the associated respective eigenvectors. I hope it is not a problem to post this as a comment. I got a couple of Courics here last time for posting a comment in the answer site.

Mr. Arturo: Interesting approach!. This seems to connect with the theory of characteristic curves in PDE's(who knows if it can be generalized to dimensions higher than 1), which are curves along which a PDE becomes an ODE, i.e., as you so brilliantly said, curves along which the PDE decouples.

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yes, the method of characteristics can be generalized to dimensions higher than 1. In general the method of characteristics for partial differential equations can be had for arbitrary first-order quasilinear scalar PDEs defined on any smooth manifold. It however will not necessarily work for systems of PDEs or higher order PDEs. –  Willie Wong Feb 23 '11 at 11:03
    
(It can in fact also be extended to fully nonlinear first order scalar PDEs, but changing the point of view from vector fields to Monge cones is a bit hard to visualize at first.) –  Willie Wong Feb 23 '11 at 11:06
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I would like to direct you to an answer that I posted here: Importance of eigenvalues

I feel it is a nice example to motivate students who ask this question, in fact I wish it were asked more often. Personally, I hold such students in very high regard.

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