I'm trying to rearrange the following equation to get only Q on the LHS:
$$ A = -Q + t(Q M + M^T Q - kQ) $$
Q is a symmetric matrix. A is a known symmetric matrix, M is an unknown non-symmetric matrix. t, k are known constants.
This seemed straightforward to me at first given all terms share a common Q, but non-commutativity of the terms is tripping me up. I could inject identity matrices and vectorize both sides, but I'd rather keep this in matrix form if possible.
We can write $$ \begin{align} A &= -Q + t (QM + M^T Q - k Q) \\ &= - Q + tQM + tM^T Q - ktQ \\ &= - Q + \left( t M^T Q \right)^T + \left( tM^T Q \right) - ktQ \\ &= -(1 +kt)Q + \left( t M^T Q \right)^T + \left( tM^T Q \right), \end{align} $$ which clearly is a symmetric matrix.
I don't think we can do much more than this to factor out $Q$.