Given K s.t. $\frac{K+K^*}{2}=L^*L+I$ and $\frac{K-K^*}{2i}=\frac{L+L^*}{2}$, find $L$.

Question

Given $K\in\mathbb{C}^{n\times n}$, and also that

$$\frac{K+K^*}{2}=L^*L+I\quad\text{and}\quad\frac{K-K^*}{2i}=\frac{L+L^*}{2},$$

where $L\in\mathbb{C}^{n\times n}$ is unknown matrix, $i=\sqrt{-1}$ and $I$ is $n\times n$ identity matrix.

Find $\frac{L-L^*}{2i}$ in terms of $K$.

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Given $A$, lets define $A_H=\frac{A+A^*}{2}$ and $A_S=\frac{A-A^*}{2i}$.

Then we get

1. $K=K_H+iK_S=(L^*L+I)+iL_H$,
2. $K_S=L_H$,
3. We need to find $L_S$ in terms of $K$.

Since \begin{align} K_H&=L^*L+I\\&=(L_H+iL_S)^*(L_H+iL_S)+I\\&=L_H^2+iL_HL_S-iL_SL_H+L_S^2+I\\&=K_S^2+iK_SL_S-iL_SK_S+L_S^2+I, \end{align}

we get $K_H=K_S^2+iK_SL_S-iL_SK_S+L_S^2+I.$ I stuck here, couldn't retrieve $L_S$.