Why were eigenvalues and eigenvectors defined?

[Tosh5457]

I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?

[micromass]

Wiki knows all: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
Check under section "history"

[Tosh5457]

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?

[AlephZero]

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.

[lavinia]

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.