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!



Specific quadratic matrix equation
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".


