Transformation of the gaussian random variables

Question

Solve the following problem:

a. Let Xr and Xi be statistically independent zero-mean Gaussian random variables with identical variance. Show that a (rotational) transformation of the form

Y_r + jY_i = (X_r + jX_i)e^jΦ

Results in another pair (Yr, Yi) of Gaussian random variables that have the same joint PDF as the pair (Xr, Xi).

b. Note that

[Y_r]  = A  [x_r]
[Y_i]          [X_i]

Where A is a 2 × 2 matrix. As a generalization of the two-dimensional transformation of the Gaussian random variables considered in (a), what property must the linear transformation Asatisfy if the PDFs for X and Y, where Y = AX, X = (X1X2 ··· Xn), and Y = (Y1Y2 ··· Yn) are identical?