I am working on the proof part in Definition section of https://en.wikipedia.org/wiki/Matrix_normal_distribution. I can understand what s going on, except for the last part how inv(V(kron)U) becomes inv(U) for multivariate distribution. is there any source you can help me with how to get multivariate pdf from matrix variate?
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The matrix-variate normal (MxVN) distribution is just a multivariate normal (MVN) distribution with a covariance matrix that is a Kronecker product. The following equivalence holds $$ X \sim N_{p,q}(M,U,V) \iff vec(X) \sim N_{pq}(vec(M),V\otimes U). $$ From this equivalence it is straight forward to derive the pdf of the matrix normal distribution as $$ |2\pi V\otimes U|^{-1/2}exp(-0.5(vec(X-M)'(V\otimes U)^{-1}vec(X-M)). $$ It seems to me that you are trying to show the following for $Z = X-M$: $$ (vec Z)'(V\otimes U)^{-1} (vec Z) = tr[ZU^{-1}Z'V^{-1}]. $$ This follows immediately from three facts. The first fact is that the inner product between two matrices, is also the vector inner product of their vectorizations $$ (vec X)' (vec X) = tr(XX'). $$ The second fact is that $vec( AXB') = (B\otimes A)vec (X) $. One last fact that is also useful is that $(U\otimes V)^{-1} = U^{-1}\otimes V^{-1}$. These 3 facts should convince you that the identity is true.