Notation: to consider a 'Gaussian' approximation of the d-dimensional unit sphere, by considering a Gaussian random vector X N (0, I/d), where I is the d d identity matrix. This means that for large input dimension d, a draw X from the N (0, I/d) Gaussian distribution will concentrate to the unit sphere X21.

LetARdd be any unitary matrix, ieATA=AAT=I Show that the pdf ofXN(0,I/d) and the pdf of X~=AXare the same, i.e they do not depend on the choice of A.

Thank you!