▁mult ivari ate ▁normal ▁distribution ▁to ▁matrix - valu ed ▁random ▁variables . ▁ ▁Definition ▁ ▁The ▁probability ▁density ▁function ▁for ▁the ▁random ▁matrix ▁X ▁( n   ×   p ) ▁that ▁ ▁follows ▁the ▁matrix ▁normal ▁distribution ▁ ▁has ▁the ▁form : ▁ ▁where ▁ ▁denotes ▁trace ▁and ▁M ▁is ▁n   ×   p , ▁U ▁is ▁n   ×   n ▁and ▁V ▁is ▁p   ×   p . ▁ ▁The ▁matrix ▁normal ▁is ▁related ▁to ▁the ▁mult ivari ate ▁normal ▁distribution ▁in ▁the ▁following ▁way : ▁ ▁if ▁and ▁only ▁if ▁ ▁where ▁ ▁denotes ▁the ▁Kr one cker ▁product ▁and ▁ ▁denotes ▁the ▁vector ization ▁of ▁. ▁ ▁Proof ▁The ▁equivalence ▁between ▁the ▁above ▁matrix ▁normal ▁and ▁mult ivari ate ▁normal ▁density ▁functions ▁can ▁be ▁shown ▁using ▁several ▁properties ▁of ▁the ▁trace ▁and ▁Kr one cker ▁product , ▁as ▁follows . ▁We ▁start ▁with ▁the ▁argument ▁of ▁the ▁exponent ▁of ▁the ▁matrix ▁normal ▁PDF : ▁▁▁ ▁which ▁is ▁the ▁argument ▁of ▁the ▁exponent ▁of ▁the ▁mult ivari ate ▁normal ▁PDF . ▁The ▁proof ▁is ▁completed ▁by ▁using ▁the ▁determin ant ▁property : ▁ ▁Properties ▁If ▁, ▁then ▁we ▁have ▁the ▁following ▁properties : ▁ ▁Ex pected ▁values ▁The ▁mean , ▁or ▁expected ▁value ▁is : ▁ ▁and ▁we ▁have ▁the ▁following ▁second - order ▁expect ations : ▁ ▁where ▁ ▁denotes ▁trace . ▁ ▁More ▁generally , ▁for ▁appropri ately ▁dimension ed ▁matrices ▁A , B , C : ▁ ▁Trans formation ▁Trans pose ▁transform :