Jordan normal form, motivation

[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 [itex] (1, 0, 0...)^{T}[/itex] 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 [itex] (1, 0, 0...)^{T}[/itex] 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

 

[paranormal]

The Jordan normal form is a powerful tool in linear algebra that allows us to simplify and analyze complex matrices. It is a way to represent a non-diagonalizable matrix in a simpler form by breaking it down into a diagonal matrix with some additional "blocks" of numbers. This form is useful because it reveals important information about the matrix, such as its eigenvalues and eigenvectors, and can help us solve systems of linear equations and understand the behavior of dynamical systems.

In physics, matrices often arise in the study of systems that evolve over time, such as quantum mechanical systems or systems described by differential equations. The Jordan normal form can be used to analyze the stability and behavior of these systems, and to find solutions to the equations that describe them. It can also be used to simplify calculations and make predictions about the behavior of these systems.

For example, in quantum mechanics, the Jordan normal form can be used to find the energy levels and states of a quantum system, and to determine the probabilities of different outcomes in measurements. In classical mechanics, it can be used to study the stability of dynamical systems, such as the behavior of a pendulum or a planetary orbit.

Overall, the Jordan normal form is a valuable tool for physicists because it allows us to better understand and solve complex systems that arise in our research. It is an important concept to grasp in order to fully utilize its applications in various areas of physics. I encourage you to continue studying and practicing with this topic in your course, as it will prove to be a valuable tool in your future studies and research.