gR \left( \h, \a \right) = \bigR \left( \S\h, \S\a \right)$ always holds. Therefore, $\S{\a^\star}$ is optimal for $\S\h$ with the same power constraint $P$.

(Nonnegative Ordered Vector) A vector $\h$ is said to be nonnegative ordered if its elements are nonnegative and in nondecreasing order according to their indices.

\[lemma:h2hBar\] For any vector $\h$, there exists a signed permutation matrix $\S$ such that $\S\h$ is nonnegative ordered.

To find such an $\S$ in Lemma \[lemma:h2hBar\], we can simply choose $\S=\P\T$, where $\T$ is a signature matrix that converts all the elements in $\h$ to nonnegative, and $\P$ is a permutation matrix that sorts the elements in $\T\h$ in nondecreasing order.

With Theorem \[theorem:ProblemTransformation\] and Lemma \[lemma:h2hBar\], for any channel vector $\h$, we can first find a signed permuation matrix $\S$ and transform $\h$ to the nonnegative ordered ${\bar{\h}}=\S\h$, then obtain the optimal coefficient vector ${{\bar{\a}}^\star}$ for ${\bar{\h}}$, and finally recover the desired optimal coefficient vector ${\a^\star}=\S^{-1}{{\bar{\a}}^\star}$ for $\h$. In this way, it suffices to focus on solving the problem in for nonnegative ordered channel vectors ${\bar{\h}}$.

\[remark:Transformation\] In implementation, there is no need to use the signed permutation matrix $\S$. It is merely necessary to: 1) record the sign of the elements in $\h$ with a vector $\bst=\fsign{\h}$, and 2) sort $\boldsymbol{\hbar}=\fabs{\h}$ in ascending order as ${\bar{\h}}$ and record the original indices of the elements with a vector $\p$ such that ${\bar{\h}}(\ell)=\boldsymbol{\hbar}(\p(\ell))$, $\ell=1,2,\cdots,L$. After ${{\bar{\a}}^\star}$ for ${\bar{\h}}$ is obtained, ${\a^\star}$ for $\h$ can be recovered with ${\a^\star}(\p(\ell))=\bst(\p(\ell)){{\bar{\a}}^\star}(\ell)$, $\ell=1,2,\cdots,L$.

Given a channel vector as $\h=[-1.9,0.1,1.1]^T$, then $\bst=[-1,1,1]^T$, $\fabs{\h}=[1.9,0.1,1.1]^T$, ${\bar{\h}}=[0.1,1.1,1.9]^T$, and $\p=[2,3,1]^T$. If for certain power $P$, ${{\bar{\a}}^\