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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

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I want to know the procedure to deal with this proposition – sword May 22 '13 at 11:32
up vote 0 down vote accepted

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

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