Why were eigenvalues and eigenvectors defined?

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Discussion Overview

The discussion centers on the historical emergence and necessity of eigenvalues and eigenvectors, exploring their origins in linear algebra, quadratic forms, and differential equations. Participants seek to understand the foundational reasons for their definition and application in various contexts.

Discussion Character

  • Exploratory
  • Historical

Main Points Raised

  • Some participants note that eigenvalues and eigenvectors are often introduced in linear algebra but have historical roots in the study of quadratic forms and differential equations.
  • One participant mentions Euler's work on rotational motion and the significance of principal axes, which are linked to eigenvectors of the inertia matrix.
  • Another participant highlights Lagrange's realization of principal axes as eigenvectors and Cauchy's contributions to the classification of quadric surfaces and the terminology surrounding eigenvalues.
  • A participant provides an example involving the bending of a cantilever beam to illustrate the importance of principal axes and their relationship to eigenvalues and vectors.
  • There is a mention of the role of eigenvectors in the theory of linear ordinary differential equations (ODEs), where they define a basis for constructing other solutions.

Areas of Agreement / Disagreement

Participants express curiosity about the historical context and applications of eigenvalues and eigenvectors, but there is no consensus on the specific details of their emergence or necessity. Multiple perspectives on their historical significance and applications remain present.

Contextual Notes

Some participants express difficulty in understanding the historical development and applications of eigenvalues and eigenvectors, indicating potential gaps in knowledge or assumptions about prior familiarity with the topic.

Tosh5457
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I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?
 
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micromass said:
Wiki knows all: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
Check under section "history"

I tried to understand that before asking here, but I didn't...

Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.
Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.[11] In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.[12] Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.[13]

How exactly did they arise from the study of quadratic forms and differential equations?
 
As a simple example of why principal axes are important, consider bending of a cantilever beam with a rectangular cross section.

If you apply a force in the direction of one of the principal axes, the beam bends in the same direction as the force. The stiffness (Force / displacement) will be different for the two principal axes, depending on the relative width and depth of the beam (I = bd3/12 in one direction and b3d/12 in the other.)

If you apply a force at an angle to the principal directions, the beam does NOT bend in the same direction as the force. You can find the direction by resolving the force into components in the principal directions, finding the corresponding components of displacement, and combining them.

How all that relates to the eigenvalues and vectors of the 2x2 inertia matrix for the cross section of the beam should be fairly obvious.
 
Tosh5457 said:
I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?

The Wikipedia article is worth looking at.

In the theory of linear ODEs, eigen vectors define a basis from which all other solutions are linear combinations.
 

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