How can I transform the following proposition that is gotten in $real$ space into the corresponding one used in the $complex$ space,i.e.,$A\in C^{n\times n},x=(x\_1,...,x\_n)\in C^n$ ?


suppose that $\lVert Ax \rVert^2= x^T(A^TA)x,\ \lVert\Sigma\rVert\_p$ denotes the spectral (operator) norm of a matrix $\Sigma$


Proposition: Let $A\in R^{n\times n}$ be a matrix, and let $\Sigma:=A^TA$. Let $x=(x\_1,...,x\_n)\in R^n$ be an isotropic multivariate Gaussian random vector with mean zero. For all $t>0$,


$$\text{Pr}\left\{\lVert Ax \rVert^2 > \text{tr}(\Sigma) +2\sqrt{\text{tr}(\Sigma^2)t}+2\lVert\Sigma\rVert\_p t \right\}\leq\exp(-t).
$$


Actually, this proposition comes from the paper [link text](http://cseweb.ucsd.edu/~djhsu/papers/quadratic-ecp.pdf)
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What is the corresponding version in the complex space of this proposition got in the real space real
The answer should be
$$\text{Pr}\left\{\lVert Ax \rVert^2 > \text{tr}(\Sigma) +\sqrt{2\text{tr}(\Sigma^2)t}+\lVert\Sigma\rVert\_p t \right\}\leq\exp(-t).
$$


