I'm a bit stumped by the exponential family representation of a multi-variate Gaussian distribution. Basically, the exponential form is a generic form for a large class of probability distributions. The standard form is
f_X(x) = \exp[\theta' T(x) + F(\theta)]
where $\theta$ is a set of parameters (based on $\mu$ and $\Sigma$), $T(x)$ is a vector of sufficient statistics, and $F$ is a function of the parameters that ensures the distribution is a pdf, i.e., sums to one. For more information on this form, see http://www.cs.columbia.edu/~jebara/4771/tutorials/lecture12.pdf, http://en.wikipedia.org/wiki/Exponential_family, etc.
The "conversion" for a multi-variate Gaussian distribution to exponential family form is listed as
\theta = [\Sigma^{-1}\mu, -\frac{1}{2}\Sigma^{-1}]' T(x) = [x, x x']'
but this is confusing because the outer product x x' is a matrix and $-\frac{1}{2}\Sigma^{-1}$ is also a matrix. Thus, it seems the product between $\theta$ and $T(x)$ should result in a scalar "entry" and a matrix "entry". Obviously, this expression needs to evaluate to a scalar.
The inner product works fine in the scalar case, and I understand this conversion is computed by manipulating to the quadratic form x'Ax + b'x. Still, it seems that I am completely missing something here. Thanks for your help.