▁ ▁Linear ▁transform : ▁let ▁D ▁( r - by - n ), ▁be ▁of ▁full ▁rank ▁r ▁ ≤ ▁n ▁and ▁C ▁( p - by - s ), ▁be ▁of ▁full ▁rank ▁s ▁ ≤ ▁p , ▁then : ▁ ▁Example ▁Let ' s ▁imagine ▁a ▁sample ▁of ▁n ▁independent ▁p - dimensional ▁random ▁variables ▁ident ically ▁distributed ▁according ▁to ▁a ▁mult ivari ate ▁normal ▁distribution : ▁. ▁When ▁defining ▁the ▁n   ×   p ▁matrix ▁ ▁for ▁which ▁the ▁it h ▁row ▁is ▁, ▁we ▁obtain : ▁ ▁where ▁each ▁row ▁of ▁ ▁is ▁equal ▁to ▁, ▁that ▁is ▁, ▁ ▁is ▁the ▁n   ×   n ▁identity ▁matrix , ▁that ▁is ▁the ▁rows ▁are ▁independent , ▁and ▁. ▁ ▁Max imum ▁lik elihood ▁parameter ▁estimation ▁Given ▁k ▁ ▁matrices , ▁each ▁of ▁size ▁n   ×   p , ▁denoted ▁, ▁which ▁we ▁assume ▁have ▁been ▁sample d ▁i . i . d . ▁from ▁a ▁matrix ▁normal ▁distribution , ▁the ▁maximum ▁lik elihood ▁estimate ▁of ▁the ▁parameters ▁can ▁be ▁obtained ▁by ▁maxim izing : ▁ ▁The ▁solution ▁for ▁the ▁mean ▁has ▁a ▁closed ▁form , ▁namely ▁ ▁but ▁the ▁cov ar iance ▁parameters ▁do ▁not . ▁However , ▁these ▁parameters ▁can ▁be ▁iter atively ▁maxim ized ▁by ▁zero - ing ▁their ▁grad ients ▁at : ▁▁ ▁and ▁ ▁See ▁for ▁example ▁ ▁and ▁references ▁there in . ▁The ▁cov ar iance ▁parameters ▁are ▁non - ident ifiable ▁in ▁the ▁sense ▁that ▁for ▁any ▁scale ▁factor ,