I am having trouble with the following matrix equation:

$(K + MU)(K + MU) = U$

$K$, $M$, and $U$ are all square matrices, the values of $K$ and $M$ are known (but they don't have a particularly simple form, e.g. they are not diagonal). I would like to find a solution for $U$.

Does anyone know how this can be done? Thanks a lot!

• What do you mean by "quadratic matrix"? – Robert Israel Aug 2 '16 at 17:52
• @RobertIsrael Probably "square matrix". The problem asks to solve a certain 2nd-order (i.e. quadratic) equation in which unknown is a square matrix. I know nothing about such matters, but it appears to be a sensible question which probably has been well studied. – Boris Bukh Aug 2 '16 at 23:01
• Indeed, I meant square matrix, just edited the post. – Matze Aug 3 '16 at 11:58

Set $Y=K+MU$. You have $Y^2=U$, so $K+MY^2=K+MU=Y$, which gives an equation in $Y$ only: $$K-Y+MY^2 = 0$$ This is a widely studied equation; see for instance Higham and Kim, http://www.maths.manchester.ac.uk/~higham/narep/narep347.pdf. In particular, there are several solutions, whose eigenvalues are $n$ out of the $2n$ (or fewer) solutions of $\det(K+Mx^2-x)=0$.
Once you have $Y$, $U=Y^2$, of course.
Since you call them $M$ and $K$, one may guess that they are a positive-definite mass matrix and a symmetric stiffness matrix, which is the context in which these problems are most often studied. See also "quadratic eigenvalue problem".