I have a real unitary matrix $D$ multiplied with a real vector $x$ resulting $y$, $(y = D \times x)$. I need to set the sign of each element of $x$ subject to reduce $\max|y|$, where $|.|$ is the absolute value.
My concern, can I do that process without iteratively multiplying $D$ with $x$? I mean without performing many multiplications of $D$ and $x$ with different signs and then selecting the minimum value.
Okay, so if I get you correct you need to minimize the absolute value of $y$ in a cartesian setting i.e. the vectors $x, y$ are elements of $\mathbb{R}^n$ and the unitary matrix $D \in \mathbb{R}^{n \times n}$ although the latter is not so precise. I will asume the matrix $p$-norm with in this particular case $p=2$ which naturally develops from the understanding of the Eucidian norm in $\mathbb{R}^n$. Now also it is known that this norm is invariant under unitary operations, meaning $||y||_2 = ||D \cdot x||_2 = ||D||_2\cdot ||x||_2$. Hence we actually have to choose the signs of $x$ such that our familiar $||x||_2 = \sqrt{\sum_{i \geq 1}^{n} x_i^2}$ gets minimized. Notice how the choice of signs of any of the terms $x_i^2$ does not have any effect on this value, because of the squaring operation. So we conclude that there is no optimal choice, rather any choice is as optimal as the next one.
Please try to be more precise in asking questions as many of parts of the question are not entirely clear. For example what is the absolute value of a vector, this is probably a norm but there are many norms and you should be precise about which one. Hopefully I helped you.