Simplifying a matrix algebra equation (revised)

[tomizzo]

Homework Statement

Verify that matrix equation can simplify to desired form

Relevant Equations

Below

I have a matrix equation (left side) that needs to be formatted into another form (right side). I've simplified the left side as much as I could but can't seem to get it to the match the right side. I am unsure if my matrix algebra skills are lacking or if I somehow messed up the starting equation making it impossible to yield the desired form.

Moderator note: The original attached image has been deleted. The following is the corrected version.
$K = (Q - QH^T(R + HQH^T)^{-1}HQ)H^TR^{-1} \Rightarrow K = QH^T(HQH^T + R)^{-1}$
Note that Q and R are invertible. H is not.

 

[tnich]

$K = (Q - QH^T(R + HQH^T)^{-1}HQ)H^TR^{-1} \Rightarrow K = QH^T(HQH^T + R)^{-1}$

It's just an algebra trick. Try adding $-QH^T(R + HQH^T)^{-1}RR^{-1} + QH^T(R + HQH^T)^{-1}RR^{-1}$ to $K$.