Error update in matrix Kalman Filter Kalman filter represents a best pathway between prediction and measurement.  Since in its simplest form the model is purely Gaussian, it achieves this by creating a product Gaussian by multiplying together the pdfs of the prediction and measurement (from which the new mean and variance can be extracted).  In one dimension, the fused ${\sigma}_{fused}^2$ is given by:  $$ {\sigma}_{fused}^2={1\over{1/{\sigma}_{pred}^2+1/{\sigma}_{meas}^2}}$$
and I can see this formula in the one dimension error update.  In the multidimensional case, it becomes:  $${\Sigma}_{fused}=({\Sigma}_{pred}^{-1}+{\Sigma}_{meas}^{-1})^{-1}$$ There is probably some kind of algebraic simplification or other sleight of hand occurring but I cannot seem to find an expression that looks like this anywhere in the multidimensional update equations.  Perhaps there is someone out there that can shed light on what I am missing.  Thank you.
Short Answer:
The updated covariance matrix in the Kalman filter can be written concisely in a single equation as follows:
$$ P_{k+1} = \bigr( (FP_k F + Q)^{-1} + H^T R^{-1} H \bigr)^{-1}$$
where $P_{k+1}$ is the fused or updated covariance, $P_{pred} = FP_k F + Q$ is the predicted covariance, and $P_{meas} = \bigr( H^T R^{-1} H \bigr)^{-1}$ is the measurement information after projecting on the state space. If you have $H=I$ where $I$ is the identity matrix, then $H^T R^{-1} H = R^{-1}$.
This is equivalent to the equation you provided in the case that $H=I$. In other words, relating to your notation:
$$ P_{fused} = P_{k+1} = \bigr( (FP_k F + Q)^{-1} + H^T R^{-1} H \bigr)^{-1} = \bigr( P_{pred}^{-1} + P_{meas}^{-1} \bigr)^{-1}$$
Long Answer:
I am not sure why the downvotes (especially with no comments clarifying the reason...), but I will provide some additional information that will hopefully help clarify my answer.
Let $F$ be the process model, $H$ be the measurement model, $Q$ be the process covariance, and $R$ be the measurement covariance. See Wikipedia article for details on what these matrices represent. Furthermore, let $P_k^{-}$ and $P_k^{+}$ denote the covariance matrix at time step $k$ for the predict step (i.e., before acquiring a measurement) and the update step (i.e., after acquiring a measurement), respectively.
The covariance matrix computed by the predict step of the Kalman filter assuming additive noise is given by
$$P_{k+1}^{-} = FP_k^{+}F + Q$$
The covariance matrix computed by the update step of the Kalman filter assuming additive noise is given by
$$P_{k+1}^{+} = (I - P_{k+1}^{-} H^T (H P_{k+1}^{-} H^T + R)^{-1} H)P_{k+1}^{-} $$
where $I$ is the identity.
Using the Woodbury Matrix Identity as mentioned in the comments by @KwinvanderVeen, the information form of the updated covariance matrix is given by
$$ (P_{k+1}^+)^{-1} = (P_{k+1}^{-1})^{-1} + H^T R^{-1} H $$
where more details of this form referred to as the information filter can be found on the Wikipedia article. As an exercise, I suggestion attempting to apply the Woodbury Matrix Identity to obtain this equation.
Taking the inverse of both sides and substituting $P_{k+1}^{-1}$ we have the following:
$$ P_{k+1}^{+} = \bigr( (FP_k^{-} F + Q)^{-1} + H^T R^{-1} H \bigr)^{-1},$$
which is equivalent to the form you describe, which using your notation is given as
$$ P_{fused} = P_{k+1}^{+} = \bigr( (FP_k^{-} F + Q)^{-1} + H^T R^{-1} H \bigr)^{-1} = \bigr( P_{pred}^{-1} + P_{meas}^{-1} \bigr)^{-1}$$
Short Answer:
The updated covariance matrix in the Kalman filter can be written concisely in a single equation as follows:
$$ P_{k+1} = \bigr( (FP_k F + Q)^{-1} + H^T R^{-1} H \bigr)^{-1}$$
where $P_{k+1}$ is the fused or updated covariance, $P_{pred} = FP_k F + Q$ is the predicted covariance, and $P_{meas} = \bigr( H^T R^{-1} H \bigr)^{-1}$ is the measurement information after projecting on the state space. If you have $H=I$ where $I$ is the identity matrix, then $H^T R^{-1} H = R^{-1}$.
This is equivalent to the equation you provided in the case that $H=I$. In other words, relating to your notation:
$$ P_{fused} = P_{k+1} = \bigr( (FP_k F + Q)^{-1} + H^T R^{-1} H \bigr)^{-1} = \bigr( P_{pred}^{-1} + P_{meas}^{-1} \bigr)^{-1}$$
Long Answer:
I am not sure why the downvotes (especially with no comments clarifying the reason...), but I will provide some additional information that will hopefully help clarify my answer.
Let $F$ be the process model, $H$ be the measurement model, $Q$ be the process covariance, and $R$ be the measurement covariance. See Wikipedia article for details on what these matrices represent. Furthermore, let $P_k^{-}$ and $P_k^{+}$ denote the covariance matrix at time step $k$ for the predict step (i.e., before acquiring a measurement) and the update step (i.e., after acquiring a measurement), respectively.
The covariance matrix computed by the predict step of the Kalman filter assuming additive noise is given by
$$P_{k+1}^{-} = FP_k^{+}F + Q$$
The covariance matrix computed by the update step of the Kalman filter assuming additive noise is given by
$$P_{k+1}^{+} = (I - P_{k+1}^{-} H^T (H P_{k+1}^{-} H^T + R)^{-1} H)P_{k+1}^{-} $$
where $I$ is the identity.
Using the Woodbury Matrix Identity as mentioned in the comments by @KwinvanderVeen, the information form of the updated covariance matrix is given by
$$ (P_{k+1}^+)^{-1} = (P_{k+1}^{-1})^{-1} + H^T R^{-1} H $$
where more details of this form referred to as the information filter can be found on the Wikipedia article. As an exercise, I suggestion attempting to apply the Woodbury Matrix Identity to obtain this equation.
Taking the inverse of both sides and substituting $P_{k+1}^{-1}$ we have the following:
$$ P_{k+1}^{+} = \bigr( (FP_k^{-} F + Q)^{-1} + H^T R^{-1} H \bigr)^{-1},$$
which is equivalent to the form you describe, which using your notation is given as
$$ P_{fused} = P_{k+1}^{+} = \bigr( (FP_k^{-} F + Q)^{-1} + H^T R^{-1} H \bigr)^{-1} = \bigr( P_{pred}^{-1} + P_{meas}^{-1} \bigr)^{-1}$$