Solving a matrix equation involving transpose conjugates

Question

How to solve the following matrix equation involving transpose conjugates: $X+X^H+aX^HX=S$, where $X$ is the variable complex matrix, $a$ is a real scalar and $S$ is a Hermitian positive semi-definite matrix? Is there any closed-form solution? The operator $.^H$ indicated the Hermitian operation (transpose conjugate).

Answer 1

When $a=0$, the general solution is given by $X=\frac{S}{2}+iH$, where $H$ is an arbitrary Hermitian matrix.

When $a>0$, the equation can be rewritten as $\left(\sqrt{a}X+\frac{I}{\sqrt{a}}\right)^H\left(\sqrt{a}X+\frac{I}{\sqrt{a}}\right)=S+\frac{I}{a}$. Therefore the general solution is given by $X=\frac{1}{\sqrt{a}}\left[U\left(S+\frac{I}{a}\right)^{1/2}-\frac{I}{\sqrt{a}}\right]$ where $U$ is an arbitrary unitary matrix.

When $a<0$, the equation can be rewritten as $\left(\sqrt{-a}X-\frac{I}{\sqrt{-a}}\right)^H\left(\sqrt{-a}X-\frac{I}{\sqrt{-a}}\right)=-S-\frac{I}{a}$. Therefore the equation is solvable if and only if $I+aS$ is positive semidefinite. If this is the case, the general solution is given by $X=\frac{1}{\sqrt{-a}}\left[U\left(-S-\frac{I}{a}\right)^{1/2}+\frac{I}{\sqrt{-a}}\right]$, where $U$ is an arbitrary unitary matrix.