PDF of Matrix Transformation

[alpines4]

Homework Statement

Define X to be an n-vector of jointly continuous random variables X1, ..., Xn with joint pdf f(x) mapping R^n to R. Let A be an invertible nxn matrix and set Y=AX+b. I want to derive the pdf of f(y) in terms of f(x), the original pdf.

Homework Equations

The Attempt at a Solution

Given a random variable and its PDF f(x), the transformation of Y=g(X) is (given that g is one to one and thus has an inverse) f(g^{-1}(y)) * g'(y). I don't know how to generalize this to a matrix, however. I assume it will be kind of similar... Any help is appreciated. I just need some tips to get started. Thank you!

 

[sunjin09]

Homework Statement

Define X to be an n-vector of jointly continuous random variables X1, ..., Xn with joint pdf f(x) mapping R^n to R. Let A be an invertible nxn matrix and set Y=AX+b. I want to derive the pdf of f(y) in terms of f(x), the original pdf.

Homework Equations

The Attempt at a Solution

Given a random variable and its PDF f(x), the transformation of Y=g(X) is (given that g is one to one and thus has an inverse) f(g^{-1}(y)) * g'(y). I don't know how to generalize this to a matrix, however. I assume it will be kind of similar... Any help is appreciated. I just need some tips to get started. Thank you!

Intuitively the pdf of Y=g(X) is given by
$$f_X(x)|dx|=f_Y(y)|dy|$$
In the case of vector x and y, dx and dy is related through the determinant of the Jacobian matrix between x and y