Graduate Eigenvalue Problem and the Calculus of Variations

[member 428835]

Hi PF!

Given $B u = \lambda A u$ where $A,B$ are linear operators (matrices) and $u$ a function (vector) to be operated on with eigenvalue $\lambda$, I read that the solution to this eigenvalue problem is equivalent to finding stationary values of $(Bu,u)$ subject to $(Au,u)=1$, where $(g,f) = \int fg$.

Can someone explain this to me, or point me in the right direction? I don't see how the two relate.

 

[Orodruin]

Did you try applying the method of Lagrange multipliers to the stationary value problem?

 

[member 428835]

Did you try applying the method of Lagrange multipliers to the stationary value problem?

Could you elaborate? I've seen something like this done before I think, where $u = \sum_i \alpha_i w_i$ where $\alpha$ is a constant and $w$ is a known trial function. By construction $(w_i,w_j)=\delta_{ij}$. Then evidently we choose $\alpha$ to minimize $(Bu,u)$ (why?) under the constraint $\sum \alpha_i^2=1$, and hence we arrive at the eigenvalue problem $Bu=\lambda u$.

 

[WWGD]

Hi PF!

Given $B u = \lambda A u$ where $A,B$ are linear operators (matrices) .

Note that Linear Operator and Matrix coincide only in finite-dimensional spaces. There is no such representation in infinite -dim spaces.

 

[StoneTemplePython]

If you're still trying to tackle the problem of finding eigenvalues of $B^{-1}A$ or $A^{-1}B$ --you posted about this a few months back as I recall, then you may want to check out this thread:

https://www.physicsforums.com/threads/eigenvalues-of-the-product-of-two-matrices.588101

 

[member 428835]

If you're still trying to tackle the problem of finding eigenvalues of $B^{-1}A$ or $A^{-1}B$ --you posted about this a few months back as I recall, then you may want to check out this thread:

https://www.physicsforums.com/threads/eigenvalues-of-the-product-of-two-matrices.588101

Yes, I definitely did ask about this a while ago. I ended up taking someone's advice on here (don't recall who it was) and used a build in function that worked great (errors were mine but shockingly also the paper's).