There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ones)

My question:
What is the most helpful intuition to get a feeling for what this "eigen" really means (in its core)? What is the connecting element? And what examples would you give to clarify this concept?

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    $\begingroup$ With the exception of eigenvalue, in most cases that I know of, eigenX means an X which is also an eigenvector for something. $\endgroup$ – Donu Arapura Jul 14 '10 at 13:00
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    $\begingroup$ As a word, "Eigen" means "belonging to oneself of itself; typical, characteristic". All instances in mathematics that I am aware of refer to the usual linear algebra situation, where a linear endomorphism of a vector space acts on a vector, or subspace, as multiplication by a fixed scalar. This seems pretty clear already. Or am I missing some subtlety? $\endgroup$ – Pete L. Clark Jul 14 '10 at 15:08
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    $\begingroup$ I agree with Donu and Pete, and so to me this seems more like once concept appearing in many different settings, rather than many things which need explaining. I would appreciate it if the author could explain what we're missing. $\endgroup$ – Yemon Choi Jul 14 '10 at 19:04
  • $\begingroup$ Also, I'm not sure that a separate "eigenvector" is needed here, or helpful in general $\endgroup$ – Yemon Choi Jul 14 '10 at 19:05
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    $\begingroup$ This uniform explanation in terms of eigenvectors fails miserably for “eigenvariable”. $\endgroup$ – Emil Jeřábek Jul 28 '11 at 11:06

When you see the word eigen, replace it with the term spectrum of an operator (see spectral theory) View the matrix as a continuous or discrete linear transform acting on a vector. Similar matrices ($B = MAM^{-1}$) represent the same transform with respect to a different base.

When you diagonalize the matrix, you are actually trying to obtain an orthogonal decomposition of the transform as a linearly independent eigensystem.

  1. If there are n independent eigenvectors, you will obtain a full diagonalization of your matrix.
  2. If less than n, you have two choices. If all eigenvalues are in the ground field, you will get a Jordan decomposition. Otherwise, you have to settle with a rational canonical form.

In addition to Gilbert Strang's excellent book and lectures on Linear Algebra, I recommend browsing through Castillo's Orthogonal sets and polar methods in linear algebra. Throughout the book, the matrix is seen as a transform rather than something which must be numerically manipulated.

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One incredibly cool lecture by Prof. Gilbert Strang about Eigenvalues and Eigenvectors.

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As explained by SandeepJ, "eigen..." is related to the spectrum of something. In particular, when one calculates the eigenvalues and their corresponding eigenvectors from A.x = lambda*x, adding the same vector x times a constant to both sides of the equation (say tau, i.e., one adds tau*x), only the eigenvalues get shifted by an amount equivalent to tau, but not the eigenvectors. The corollary that one can get from this statement, is that "eigen..." is about locating some characteristic functions or operators, which get unchanged under the shift of a constant, or something similar. We can state it in another way: in the process of studying "eigen..." we want to find some functions on the spectrum of some sort of functional, which remain invariant under certain types of transformation. This last phrase may sound pretty vague, but the thing is that "eigen..." is not something for which we can always define a straightforward algorithm.

The place which I liked the most for answering this question of yours is "Numerical Recipes in Fortran. The Art of Scientific Computing" (William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery)

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