Solve a matrix equation involving a Frobenius norm

Question

I am trying to solve for X an equation of the form : $$X\Vert X\Vert_F + A X + B = 0$$

where $X$, $A$ and $B$ are square matrices. Is there even an analytical solution ? Any hint is greatly appreciated :-)

Answer 1

I'm afraid there is not any closed-form solution formula, but the problem can be solved as follows. Denote the spectrum of $A$ by $\sigma(A)$ and denote the Moore-Penrose pseudo-inverse of a matrix $M$ by $M^+$. The equation in question has a solution $X$ if and only if the system of equations $(A+tI)X=-B$ and $\|X\|_F=t$ has a solution $(X,t)$. Hence it is solvable if and only if

1. The rational function $f(t)=\|(A+tI)^{-1}B\|_F^2-t^2$ has a real zero on $[0,\infty)\setminus\sigma(A)$ (and in this case, $X=-(A+tI)^{-1}B$ is a solution), or
2. $A$ has a real nonpositive eigenvalue $-t\le0$ such that $(A+tI)X=-B$ is solvable and $\|(A+tI)^+B\|_F\le t$. In this case, the solutions are given by $X=Y-(A+tI)^+B$, where $Y$ is any matrix satisfying $(A+tI)Y=0$ (i.e. columns of $Y$ are eigenvectors of $A$ corresponding to the eigenvalue $-t$) and $\|Y\|_F=\sqrt{t^2-\|(A+tI)^+B\|_F^2}$.