suppose $$ x_t\sim Normal(x\mid\mu,\Sigma) $$ where $x_t\in R^{N}$and $$ x_t=Ax_{t-1}+v_t (Eq.1) $$ where $$ v_t\sim Normal(v_t\mid 0,Q^{-1}) $$ and $Q$ is a $N\times N$ random matrix $$ Q\sim Wishart (w,\Psi) $$ where $w$ is a real number and $\Psi\in R^{N\times N}$ is a semi-positive matrix of real values.
we want to compute $$ E[(x_t-Ax_{t-1})^TQ(x_t-Ax_{t-1})] $$
what I have done is (Using Eq.1): $$ E[(x_t-Ax_{t-1})^TQ(x_t-Ax_{t-1})]=E[(Ax_{t-1}+v_t-Ax_{t-1})^TQ(Ax_{t-1}+v_t-Ax_{t-1})] $$ $$ =E[v_t^TQv_t]=\sum_{n=1}^N\sum_{n^\prime=1}^N E[v_{t_n}^TQ_{nn^\prime}v_{t_{n^\prime}}] $$ $$ =\sum_{n=1}^N\sum_{n^\prime=1}^N E[E[v_{t_n}Q_{nn^\prime}v_{t_{n^\prime}}\mid Q_{nn^\prime}]]=\sum_{n=1}^N\sum_{n^\prime=1}^N E[Q_{nn^\prime}E[v_{t_n}v_{t_{n^\prime}}\mid Q_{nn^\prime}]] $$ $$ =\sum_{n=1}^N\sum_{n^\prime=1}^N E[Q_{nn^\prime}Q^*_{nn^\prime}] $$ where $Q^*$ is the inverse matrix of $Q$.
How can I proceed with the equation?