Jacobian of the linear transform Y = AX

In summary, the conversation discusses finding the Jacobian of a transformation represented by the equation Y = AX = g(X), where X and Y are elements of R^n and A is an nxn matrix. It is determined that the Jacobian is equal to the determinant of A, regardless of whether it is represented as a matrix or a vector.
  • #1
Legendre
62
0

Homework Statement



Y = AX = g(X)

Where X,Y are elements of R^n and A is a nxn matrix.

What is the Jacobian of this transformation, Jg(x)?


Homework Equations



N.A.

The Attempt at a Solution



Well, I know what to do in the non-matrix case. For example...

U = g(x,y)
V = h(x,y)

The transformation can be seen as a vector valued function f(x,y) = (g(x,y),h(x,y)). So the jacobian of this transform, Jf(x,y) = the determinant of a matrix with row 1 = [dg/dx , dh/dx] and row 2 = [dg/dy, dh/dy].

So Jf(x,y) = (dg/dx)(dh/dy) - (dg/dy)(dh/dx).

But what do I do in the matrix case? I know g(X) can be seen as a vector with functions as entries but does this help?

Y = AX = g(X) = (g1(X),g2(X),...,gn(X))

Thanks!
 
Physics news on Phys.org
  • #2
If you mean that A is a constant matrix, then the Jacobian is just the determinant 0f A.
 
  • #3
I agree with HallsofIvy:

It don't think it matters to have a matrix, I do consider it as a vector of R^n as you do Legendre. Then I write the matrix of Jacobi of this function and find A; so it's jacobian is the determinant of A.
 
  • #4
Thanks guys. I wrote Ax, for a constant martix A, as a linear combination of its columns, then deduce that each of the gi(X) is a linear combination of the entries in the ith row of A. Then the jacobian is the determinant of A transposed, which is equal to the determinant of A!
 

1. What is the Jacobian of a linear transform?

The Jacobian of a linear transform is a matrix that represents the rate of change of the transformed variables with respect to the original variables. It captures how small changes in the original variables affect the transformed variables.

2. How is the Jacobian calculated for a linear transform?

The Jacobian of a linear transform can be calculated by taking the partial derivatives of the transformed variables with respect to the original variables and arranging them in a matrix. The resulting matrix will have the same dimensions as the original and transformed variables.

3. What does the Jacobian tell us about a linear transform?

The Jacobian matrix tells us about the relationship between the original and transformed variables. It can help us understand how changes in the original variables affect the transformed variables and vice versa. It also provides information about the scaling and rotation of the transformed variables.

4. How is the Jacobian used in linear algebra?

In linear algebra, the Jacobian matrix is used to study the properties of linear transformations. It can help in finding the inverse of a linear transform and in solving systems of linear equations. It is also used in multivariable calculus to calculate integrals and determine convergence of series.

5. Can the Jacobian be applied to non-linear transforms?

Yes, the concept of the Jacobian can also be extended to non-linear transforms. In these cases, the Jacobian is a matrix of partial derivatives that captures the rate of change of the transformed variables with respect to the original variables. However, the calculation of the Jacobian for non-linear transforms can be more complex compared to linear transforms.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
515
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Special and General Relativity
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
764
  • Calculus and Beyond Homework Help
Replies
1
Views
886
  • Calculus and Beyond Homework Help
Replies
1
Views
980
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
5K
Replies
9
Views
2K
Back
Top