 Solution

Use the usual state-space assumption that the prior $X_0$ is independent of future process and observation noises. Suppose inductively $X_t\mid {\mathcal{Y}}_t\sim N({\hat{X}}_t,V_t)$. Prediction through the linear dynamics gives $X_{t+1}\mid {\mathcal{Y}}_t\sim N(A{\hat{X}}_t,M_t)$, where $M_t=A{V}_t{A}^T+{\Sigma }_{\epsilon }$. The pair $(X_{t+1},Y_{t+1})$ is conditionally multivariate normal, with means $A{\hat{X}}_t,CA{\hat{X}}_t$, cross covariance $M_t{C}^T$, and observation covariance $R_t={\Sigma }_{\eta }+C{M}_t{C}^T$. Applying part (a) gives the Kalman filterThis proves the inductive Gaussian assertion. The inverse requires nonsingular innovation covariance, as in the printed formula. Importantly, prediction uses $A{\hat{X}}_t$: the TeX conversion drops this $A$ in one place, whereas the original PDF includes it.