Let X be an Nn.0; E/-distributed n-dimensional random vector, and let A and B be k n and l n matrices of rank k and l , respectively. Show that AX and BX are independent if and only if AB>

D 0. Hint: Assume without loss of generality that kCl  n. In the proof of the ‘only if’ direction, verify first that the .kCl/ n matrix C WD



AB



has rank kCl , and conclude that CX has distribution NkCl .0; CC>/.