# Intuitions/connections/examples for “eigen-*”

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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With the exception of eigenvalue, in most cases that I know of, eigenX means an X which is also an eigenvector for something. –  Donu Arapura Jul 14 '10 at 13:00
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? –  Pete L. Clark Jul 14 '10 at 15:08
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. –  Yemon Choi Jul 14 '10 at 19:04
Also, I'm not sure that a separate "eigenvector" is needed here, or helpful in general –  Yemon Choi Jul 14 '10 at 19:05
This uniform explanation in terms of eigenvectors fails miserably for “eigenvariable”. –  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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–  SandeepJ Jul 14 '10 at 21:25