Let $X_1,...,X_n \sim N_p(\mu_i,\Sigma_i)$ be Multivariate Normal a.v. independent.

Show that $W = (X_1,...,X_n) \sim MN(M,\mathbb{I},\Sigma)$ where $M = [\mu_1 \mu_2...\mu_n]$ and $\mathbb{I}$ is the identity nxn.

The density of a matrix-normal variable with independent entries is $$p(W|M,\mathbb{I},\Sigma) =\frac{\exp\left(-\frac{1}{2}tr(\Sigma^{-1}(W-M)^T\mathbb{I}(W-M)\right))}{(2\pi)^{np/2}|\Sigma|^{n/2}}$$

Given that my variables $X_i$ are independent, the joint density will be the product of them, so:

$$f(W|M,\mathbb{I},\Sigma) =\prod^n_{i=1}\left\{\frac{1}{(2\pi)^{p/2}|\Sigma|^{1/2}}\exp\left[-\frac{1}{2}(X_i-\mu_i)^T\Sigma^{-1}(X_i-\mu_i)\right]\right\}$$ $$=\frac{1}{(2\pi)^{np/2}|\Sigma|^{n/2}}\exp\left[-\frac{1}{2}\sum_{i=1}^n(X_i-\mu_i)^T\Sigma^{-1}(X_i-\mu_i)\right]$$

Now, what I have to do is somehow turn the expression within the exp into a diagonal matrix in order to change the expression from a sum to the trace of a matrix. Any tips on how to do that? My professor suggested trying to use kronecker product in order to generate this matrix.

Easier ways to prove it or any books with the solution would also be really apreciated. Thanks in advance.


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