Let $z = x + \text{i}\, y$ with $x,y \in \mathbb{R}^n$ and $Z = X + \text{i}\, Y$ with $X, Y \in \mathbb{R}^{n \times n}$. How can I compute $z^H Z z$ in terms of $x, y, X, Y$ where $z^H = \bar{z}^T$ (i.e., conjugate transpose)?
I tried by hand for the real part and got something like $x^T X x – y^T Y y – y^T Y x + x^T X y$, but in attempting to validate it, I realized that I made a mistake somewhere. What's the correct formula for $z^H Z z$ in terms of $x, y, X, Y$ written as vector-matrix-vector multiplications?
Best Answer
You want the complex quantity $$Q:=\bar z^\top Z z=(x^\top-i y^\top)(X+iY)(x+iy)\ .$$ If you expand this distributively you obtain $8$ terms, four of them real, and the other four with a factor $i$. Collect each quadruple, and you are done. If $Z$ has symmetry properties untold in the question things may simplify further.