Solving u + LuL^T = v for u in the form B₁vB₂

[Skatch]

I've got something like
$$u + LuL^T = v$$
and I want to write it like
$$u = B_1 v B_2$$
for some B_1 and B_2. Assume L is nice and invertible.

Is this impossible or am I just not seeing it? This is making me feel dumb. Too much programming and its getting late.

 

[Petr Mugver]

I assume all the letters you wrote are n x n matrices...

Try multiplicating the first u on the left by LL^-1 and on the right by (LL^-1)^T, then collect terms...