I have this paragraph in the book I cannot understand (this is not the first time I encounter this thing but I usually move on).
$A^\prime TA+A^\prime QB+B^\prime UA+B^\prime RB = (A+T^{-1}QB)^\prime T(A+T^{-1}QB)+B^\prime (R-UT^{-1}Q)B$
I can do the calculus in the reverse order, but I don’t know how is someone supposed to figure out how to do it, and I don’t understand why such a procedure is called “completing the squares”.
Can someone please elaborate on that?
NOTE: prime stands for transpose and those are all matrices.