Welcome to MathOverflow

Question

1

2

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?

flag	asked Jul 14 2010 at 12:03 vonjd 2,766●3●19●37

2

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 2010 at 13:00

3

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 2010 at 15:08

1

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 2010 at 19:04

Also, I'm not sure that a separate "eigenvector" is needed here, or helpful in general – Yemon Choi Jul 14 2010 at 19:05

1

This uniform explanation in terms of eigenvectors fails miserably for “eigenvariable”. – Emil Jeřábek Jul 28 2011 at 11:06

show 2 more comments

SandeepJ · Accepted Answer · 2010-07-14 14:25:03Z UTC

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.

If there are n independent eigenvectors, you will obtain a full diagonalization of your matrix.
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.

Pratik Deoghare · Answer 2 · 2010-07-14 12:21:53Z UTC

2

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

answered Jul 14 2010 at 12:21

Pratik Deoghare
265●8

Intuitions/connections/examples for “eigen-*”

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.