The eigenvalues and eigenvectors of T

[Schwarzschild90]

Homework Statement

Homework Equations

The lattice laplacian is defined as [itex] \Delta^2 = \frac{T}{\tau} [/itex], where T is the transition matrix [tex]
\left[ \begin{array}{cccc}
-2 & 1 & 0 & 0 \\
1 & -2 & 1 & 0 \\
0 & 1 & -2 & 1 \\
0 & 0 & 1 & -2 \end{array} \right]
[/tex]

and [tex]\tau[/tex] is a time constant, which is taken = 1.

The Attempt at a Solution

 

[Ssnow]

remember that the eigenvector is a vector ##v=[O_{1}(j),O_{2}(j),O_{3}(j),O_{4}(j)]## (for the case ##4\times 4##), ##\tau=1##, so is ##v\Delta^{2}=\lambda v## ...

 

[Schwarzschild90]

remember that the eigenvector is a vector ##v=[O_{1}(j),O_{2}(j),O_{3}(j),O_{4}(j)]## (for the case ##4\times 4##), ##\tau=1##, so is ##v\Delta^{2}=\lambda v## ...

Right.

How is the lattice Laplacian commonly defined?

 

[Ssnow]

Sincerely I searched on the web and I have found only this

https://en.wikipedia.org/wiki/Discrete_Laplace_operator

this is the discrete Laplace operator and yours...

Regarding your problem (in the example that you proposed) is the same to verify the system:
##[O_{1}(j),O_{2}(j),O_{3}(j),O_{4}(j)]\left[\begin{array}{cccc}-2 & 1 & 0&0 \\ 1&-2&1&0\\0&1&-2&1\\0&0&1&-2 \end{array}\right]=[\lambda O_{1}(j),\lambda O_{2}(j),\lambda O_{3}(j),\lambda O_{4}(j)]##

that is

##-2O_{1}+O_{2}=\lambda O_{1}, O_{1}-2O_{2}+O_{3}=\lambda O_{2}, ... ## and so on ...

 

[Schwarzschild90]

Sincerely I searched on the web and I have found only this

https://en.wikipedia.org/wiki/Discrete_Laplace_operator

this is the discrete Laplace operator and yours...

Regarding your problem (in the example that you proposed) is the same to verify the system:
##[O_{1}(j),O_{2}(j),O_{3}(j),O_{4}(j)]\left[\begin{array}{cccc}-2 & 1 & 0&0 \\ 1&-2&1&0\\0&1&-2&1\\0&0&1&-2 \end{array}\right]=[\lambda O_{1}(j),\lambda O_{2}(j),\lambda O_{3}(j),\lambda O_{4}(j)]##

that is

##-2O_{1}+O_{2}=\lambda O_{1}, O_{1}-2O_{2}+O_{3}=\lambda O_{2}, ... ## and so on ...

Okay, I knew that definition of the lattice Laplacian. It's what we used in the course, but it was not defined as such.

Right.

Next step is solving the characteristic equation for the eigenvalues of the system.

 

[Ssnow]

Hi, this seems a serious article,

http://math.ucdenver.edu/~brysmith/software/Eigenvalues_of_the_discrete_laplacian_bryan_smith.pdf

I think can help you!

Ssnow

 

[Schwarzschild90]

Hi, this seems a serious article,

http://math.ucdenver.edu/~brysmith/software/Eigenvalues_of_the_discrete_laplacian_bryan_smith.pdf

I think can help you!

Ssnow

I read it through, but we haven't worked that much in-depth with eigenvalues and eigenvectors with respect to the lattice Lalplacian or used applied linear algebra sufficiently for me to easily understand that. So it's slightly above my mathematical skills, but I'll talk it through with my professor tomorrow and see what I come up with.

But from what I could garner, the eigenvalues are given simply by two formulas, one for the even-valued k and one for odd-valued k.