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Let's represent multivariate normal distribution as an exponential family:

\begin{align} f_X( x | \Theta ) = h(x)g(\Theta)\exp( \eta(\Theta) \cdot T(x) ) \end{align}

Where natural parameters:

\begin{align} \eta(\Theta) = \left[ \begin{matrix} \Sigma^{-1} \mu \\ -\frac12 \Sigma^{-1} \end{matrix} \right] \end{align}

Sufficient statistic:

\begin{align} T(x) = \left[ \begin{matrix} x \\ xx^T \end{matrix} \right] \end{align}

Does $\eta(\Theta) \cdot T(x)$ denote a dot product?

If so then we can't get a scalar value. We need some more complex function then dot product - for instance trace: $\operatorname*{Tr}(\Sigma^{-1} xx^T) = \operatorname*{Tr}(x^T\Sigma^{-1} x)= x^T\Sigma^{-1} x$

Update

What are the proper identical transformations to get:

\begin{align} \eta(\Theta) \cdot T(x) = \\ \left[ \begin{matrix} \Sigma^{-1} \mu \\ -\frac12 \Sigma^{-1} \end{matrix} \right] \cdot \left[ \begin{matrix} x \\ xx^T \end{matrix} \right] = \\ ??? = \\ -\frac12 x^T \Sigma^{-1} x + \mu^T \Sigma^{-1} x \end{align}

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However, the inner product has an alternative form $x^T\Sigma x=\sum_{i,j} x_i x_j \sigma_{ij}$ where $\Sigma=\{\sigma_{ij}\}$.

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    I can't understand how to "insert" the $\Sigma^{-1}$ between two $x$s? What kind of operator should we use here? My guess is a trace operator ... Hence, does $\eta(\Theta) \cdot T(x)$ denote any functon of $\Theta$ and $x$ ? – 2017-01-19
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    no you don't. We are trying to evaluate the bilinear form here $x^T \Sigma^{-1} x$. Expand the bilinear form term by term. – 2017-01-19
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    \begin{align} \eta(\Theta) \cdot T(x) = \\ \left[ \begin{matrix} \Sigma^{-1} \mu \\ -\frac12 \Sigma^{-1} \end{matrix} \right] \cdot \left[ \begin{matrix} x \\ xx^T \end{matrix} \right] = \\ ??? = \\ -\frac12 x^T \Sigma^{-1} x + \mu^T \Sigma^{-1} x \end{align} – 2017-01-19