When Should Physics Students Use Eigenvectors?

[imaloonru]

I'm a physics major. As such, I have come across several situations in my studies that require the use of eigenvectors and eigenvalues. Whenever I have to use this method, I've been told to. I do not have a complete understanding of eigenvectors and values and am wondering how you would spot a situation where you would need to use them.

For example, if I wanted to know when or where the rate of change of something was 0, I would take a derivative, set it equal to zero, then solve for some variable. What sort of situation would I look for (in general) that would make me say "Hey! I need to find some eigenvectors here."?

Thanks.

 

[Redbelly98]

Welcome to PF.

In general, if you have an equation of the form

Matrix x Vector = Scalar x Vector​

or

Operator x Function = Scalar x Function​

The "Scalar" is an eigenvalue that you must find.

 

[HallsofIvy]

Also you can "simplify" linear transformations, writing them as matrices in either diagonal or Jordan normal form, with eigenvalues on the diagonal, by using the eigenvectors (or if there is not a complete set of eigenvectors, the "generalized" eigenvectors) as basis for the vector space.

 

[imaloonru]

Thanks guys. This helps.