Transition matrix, Jacobian.

[HMT]

Hi:

I`m new here, can someone tell me which is the difference between the transition matrix and the Jacobian?, I did some exercises of the both topics, but How is it related?

Thanks for the attention

Sorry for my English writing, but English is not my native language.

 

[Alesak]

Hi,

by transition matrix, do you mean change-of-basis matrix?

Jacobian gives linear approximation of a smooth function between two euclidean spaces at given point.

 

[HMT]

Yes the transition matrix & change of basis matrix means the same, but for example in the change of coordinates, we need to get the Jacobian of the transformation (for the metric tensor etc), but I`m now feel confused between the matrix of the linear transformation, and the Jacobian of a transformation.

 

[lavinia]

Yes the transition matrix & change of basis matrix means the same, but for example in the change of coordinates, we need to get the Jacobian of the transformation (for the metric tensor etc), but I`m now feel confused between the matrix of the linear transformation, and the Jacobian of a transformation.

In a change of coordinates, the Jacobian gives a change of basis at each point of the domain. The basis change change varies from point to point.

 

[blue_raver22]

Hello, and welcome to the community! The transition matrix and Jacobian are both mathematical concepts used in different fields of science and engineering. While they may seem similar, they serve different purposes and have different applications.

The transition matrix is a square matrix that represents the transition probabilities between states in a system. It is often used in the field of probability and statistics to model the behavior of a system over time. The values in the matrix represent the likelihood of transitioning from one state to another, and the matrix can be used to make predictions about future states of the system.

On the other hand, the Jacobian is a matrix of partial derivatives used in calculus and linear algebra. It is used to describe the relationship between two sets of variables, and is often used in the study of systems that change over time. In this sense, it can be related to the transition matrix as it also involves the concept of change over time, but the Jacobian is more focused on the relationship between variables rather than predicting future states.

In summary, the transition matrix and Jacobian serve different purposes and are used in different contexts. However, they both play important roles in understanding and analyzing systems in various fields of science and engineering. I hope this helps clarify the difference between the two concepts. Let me know if you have any further questions.