Jacobian of the linear transform Y = AX

[Legendre]

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!

 

[HallsofIvy]

If you mean that A is a constant matrix, then the Jacobian is just the determinant 0f A.

 

[penguin007]

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.

 

[Legendre]

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!