A problem about the matrix equation $x^HAx+x^HB\bar{x}+x^TCx=0$.

Question

Let $A$, $B$, and $C$ be three $m\times m$ ($m\geq1$) complex constant matrices. For any $m$-dimensional complex-variable vector $x=(x_1,x_2,...,x_m)^T\in\mathbb{C}^m$, we denote its conjugate by $$\bar{x}=(x_1^*,x_2^*,...,x_m^*)^T$$ and conjugate transpose $$x^H=(x_1^*,x_2^*,...,x_m^*).$$ If the three matrices $A$, $B$, and $C$ satisfy $$x^HAx+x^HB\bar{x}+x^TCx=0,\quad\text{for all }x\in\mathbb{C}^m,$$ prove that $$x^HAx+x^HB\bar{x}+x^TCx =x^HAx+\frac{1}{2}x^H(B+B^T)\bar{x}+\frac{1}{2}x^T(C+C^T)x,\quad\text{for all }x\in\mathbb{C}^m.$$

Answer 1

Note that $x^H B\overline{x}$ is a scalar. Therefore $$x^H B\overline{x} = (x^H B\overline{x})^T = x^H B^T\overline{x}.$$ Same goes for $x^TCx$.