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Why were eigenvalues and eigenvectors defined? 
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#1
Sep2611, 05:28 PM

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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?



#2
Sep2611, 07:01 PM

Mentor
P: 18,290

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


#3
Sep2611, 07:46 PM

P: 239




#4
Sep2611, 09:24 PM

Engineering
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P: 7,160

Why were eigenvalues and eigenvectors defined?
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 = bd^{3}/12 in one direction and b^{3}d/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. 


#5
Sep2711, 10:03 PM

Sci Advisor
P: 1,716

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


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