Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun A vector whose direction is unchanged by a given transformation and whose magnitude is changed by a factor corresponding to that vector's eigenvalue. In quantum mechanics, the transformations involved are operators corresponding to a physical system's observables. The eigenvectors correspond to possible states of the system, and the eigenvalues to possible observed values.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun linear algebra A
vector that is not rotated under a givenlinear transformation ; a left or right eigenvector depending on context.  noun physics, engineering A right eigenvector; a nonzero vector such that, for a particular
matrix , for somescalar which is itseigenvalue and aneigenvalue of the matrix.
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
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Examples

The concept of increased positive standard of living is a valuable measure, but it is not a "eigenvector" of the economy in any sense.
Evolutionary Psychology and Economic Bias, Bryan Caplan  EconLog  Library of Economics and Liberty 2009

Î£ci fi, for some set {ci} of complex numbers, where fi, the socalled eigenvector of Q corresponding to possible value qi, is that state of S at t for which, when S is in that state, there is unit probability that Q has value qi. [
Measurement in Quantum Theory Krips, Henry 2007

The curve typically indicates a nonlinear correlation; i.e. more authenticity generates an increasingly trusting relationship. eigenvector Says:
Sure, sure. 2009

Also: “why infomercials still exist” eigenvector Says:

Hey, I think Jo must be a pigeon. eigenvector Says:
Do not feed them. 2009

The real budget in the final answer, in Greg's paper, is output from labor, socialness, and height; a three eigenvector decomposition of the general set of economies consisting of random draws according the the presented probaility distributions.
Mankiw Defends his Tall Tale, Arnold Kling  EconLog  Library of Economics and Liberty 2009

But it is indeed always the pushing that counts. eigenvector Says:
How to get unstuck. 2010

Why, just create a matrix of the links and then an eigenvector of that matrix is the answer.

Knowing how to calculate the eigenvector centrality of something is not novel.
The Volokh Conspiracy » Supreme Court Skeptical of Business Method Patents 2009

Knowing how to calculate the eigenvector centrality of something is not novel.
The Volokh Conspiracy » Supreme Court Skeptical of Business Method Patents 2009
machineplay commented on the word eigenvector
eigenvector: There was a time when I knew what this word meant  just enough to know I'd never really be able to use one.
December 7, 2006