How Is Jordan Normal Form Useful for Physicists?

[fluidistic]

Hi guys,
This isn't a homework question but it's course related. In my mathematical methods in Physics course we were introduced the Jordan normal form of a matrix.
I didn't grasp all. What I understood is that when a matrix isn't diagonalizable, it's still possible (only sometimes; depending on the given matrix), to "almost" diagonalize it.
I'd like to know what is the point of doing so, especially how can it be used for physicists and in what kind of problems such matrices can appear.
Thanks a lot!

 

[McLaren Rulez]

The main reason it helps is because you can read the eigenvectors and eigenvalues off very easily. The eigenvalues are along the diagonal. Each Jordan block yields only one eigenvector which is the column vector of the form $(1, 0, 0...)^{T}$ so you can organize your matrix into many Jordan Blocks and you have your eigenvectors too.

Maybe there are more uses but this is what strikes me as most significant.

 

[fluidistic]

The main reason it helps is because you can read the eigenvectors and eigenvalues off very easily. The eigenvalues are along the diagonal. Each Jordan block yields only one eigenvector which is the column vector of the form $(1, 0, 0...)^{T}$ so you can organize your matrix into many Jordan Blocks and you have your eigenvectors too.

Maybe there are more uses but this is what strikes me as most significant.

Thanks. I knew this but why would knowing eigenvalues be important if the matrix isn't even diagonalizable?

 

[Ray Vickson]

Hi guys,
This isn't a homework question but it's course related. In my mathematical methods in Physics course we were introduced the Jordan normal form of a matrix.
I didn't grasp all. What I understood is that when a matrix isn't diagonalizable, it's still possible (only sometimes; depending on the given matrix), to "almost" diagonalize it.
I'd like to know what is the point of doing so, especially how can it be used for physicists and in what kind of problems such matrices can appear.
Thanks a lot!

You need it in computing functions of the matrix. For example, one way to solve a linear system of constant-coefficient DEs of the form dX/dt = A*x is to write X(t) = X_0*exp(A*t), so you need the exponential of a matrix. Knowing the Jordan normal form allows you to write down the matrix exponential explicitly.

RGV