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)?
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))