The extraction matrix shown as $H_k$ below, transforms the state vector into a form that can be subtracted from the measurements vector:
$\hat{X}_k = \hat{X}_k^- + K_k ({z}_k - H_k \hat{X}_k^-)$
Where $\hat{X}_k^-$ is the projected state (from the state transition model), ${z}_k$ is the measurement vector, and $K_k$ is the Kalman gain.
Why is the filter normally formed in this way? Right now $\hat{X}_k^-$ is being transformed to the "measurement space" (for lack of a better term) and subtracted from the measurement vector, then it is transformed back into the "state space" by being multiplied by $K_k$. Why don't we just transform the measurement matrix instead so that we'd deal with only one transformation?
Is there any benefit in doing it this way?
Thanks!
EDIT: let me explain this better. I'm proposing to modify the measurement update process so it would be of this form: $\hat{X}_k = \hat{X}_k^- + L_k(G_kz_k - \hat{X}_k^-)$. I feel this would be less computationally taxing. The end result would be the same. Instead of multiplying $\hat{X}_k^-$ by $H_k$ I am multiplying $z_k$ by $G_k$. $L_k$ would be the Kalman gain after it has been modified to be compatible with these changes.