Joint Gaussian With Tensors

[John Creighto]

For vectors we can define the Joint Guasian as follows:

f_X(x_1, \dots, x_N) = \frac {1} {(2\pi)^{N/2}|\Sigma|^{1/2}} \exp \left( -\frac{1}{2} ( x - \mu)^\top \Sigma^{-1} (x - \mu) \right)

Now what if (x - \mu) is a matrix A and \Sigma is an order four covariance matrix Q between ellements of A. Can we define a higher dimensional version of the joint gausian in terms of the double dot product as follows:

f_X(x_1, \dots, x_N) = \frac {1} {(2\pi)^{N/2}|Q|^{1/2}} \exp \left( -\frac{1}{2} ( A - \bar A)^T : Q^{-1} : (A - \bar A) \right)

What I see as possible problems are perhaps (2\pi)^{N/2} should be (2\pi)^{N^2/2}

The transpose operator is ambiguous so maybe index notation is necessary, although the double dot notation seems much neater.

I understand in index notation repeated indices are summed so should I write:

[ A - \bar A]^{(i,j)} [Q^{-1}]^{(i,j,m,n)}[A - \bar A]^{m,n}

instead of:

( A - \bar A)^T : Q^{-1} : (A - \bar A)

Or maybe just get rid of the transpose operator?

Finally how well is the inverse and determinant of Q defined?

Is Q^{-1} defined so that Q:Q=I where I is rank four and is 1 on the diagonal and 0 is every where else?

Other notation issues:

is

[ A - \bar A]^{(i,j)} [Q^{-1}]^{(i,j,m,n)}[A - \bar A]^{(m,n)}

equivalent to:

[Q^{-1}]^{(i,j,m,n)}[A - \bar A]^{(m,n)}[ A - \bar A]^{(i,j)}

Seems like it should be for the case that Q is symmetric but not in general.

Maybe subscrips on indicies would be a good way to define transposes:

so [Q^{-1}]^{(i_2,j_1,m,n)} would be [Q^{-1}]^{(i,j,m,n)} with the first two indicies permuted (I'm sure this isn't the standard convention. Also note I haven't taken any courses that cover tensors so my knowledge is quite limited.