Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and unit variance. Output random vector $\mathbf{y}$ is equal to \begin{align} \mathbf{y}_{m\times 1}=\mathbf{U}_{m\times m}\mathbf{D}_{m\times n}\mathbf{V}_{n\times n}\mathbf{x}_{n\times 1}+\mathbf{z}_{m\times 1}, \end{align} where $m<n$ and $\mathbf{D}=[\mathbf{I}_m,\mathbf{0}_{m\times (n-m)}]$ in which $\mathbf{I}_m$ is the identity matrix of size $m$.

Then, what is the following mutual information maximization \begin{align} \max_{p(\mathbf{x}): \mathbb{E}[\mathbf{x}^\mathrm{H}\mathbf{x}]=1} I(\mathbf{x};\mathbf{y}), \end{align} where $(.)^\mathrm{H}$ is the Hermitian operator.

My Guess is that the compex Gaussian distribution is the maximizing distribution because it is unitary invariant.

Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and identity covaiance matrix. Output random vector $\mathbf{y}$ is equal to \begin{align} \mathbf{y}_{m\times 1}=\mathbf{U}_{m\times m}\mathbf{D}_{m\times n}\mathbf{V}_{n\times n}\mathbf{x}_{n\times 1}+\mathbf{z}_{m\times 1}, \end{align} where $m<n$ and $\mathbf{D}=[\mathbf{I}_m,\mathbf{0}_{m\times (n-m)}]$ in which $\mathbf{I}_m$ is the identity matrix of size $m$.

Then, what is the following mutual information maximization \begin{align} \max_{p(\mathbf{x}): \mathbb{E}[\mathbf{x}^\mathrm{H}\mathbf{x}]=1} I(\mathbf{x};\mathbf{y}), \end{align} where $(.)^\mathrm{H}$ is the Hermitian operator.

My Guess is that the compex Gaussian distribution is the maximizing distribution because it is unitary invariant.

Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and unit variance. Output random vector $\mathbf{y}$ is equal to \begin{align} \mathbf{y}_{m\times 1}=\mathbf{U}_{m\times m}\mathbf{D}_{m\times n}\mathbf{V}_{n\times n}\mathbf{x}_{n\times 1}+\mathbf{U}_{m\times 1}, \end{align} where $m<n$ and $\mathbf{D}=[\mathbf{I}_m,\mathbf{0}_{m\times (n-m)}]$ in which $\mathbf{I}_m$ is the identity matrix of size $m$.

Then, what is the following mutual information maximization \begin{align} \max_{p(\mathbf{x}): \mathbb{E}[\mathbf{x}^\mathrm{H}\mathbf{x}]=1} I(\mathbf{x};\mathbf{y}), \end{align} where $(.)^\mathrm{H}$ is the Hermitian operator.

My Guess is that the compex Gaussian distribution is the maximizing distribution because it is unitary invariant.

Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and unit variance. Output random vector $\mathbf{y}$ is equal to \begin{align} \mathbf{y}_{m\times 1}=\mathbf{U}_{m\times m}\mathbf{D}_{m\times n}\mathbf{V}_{n\times n}\mathbf{x}_{n\times 1}+\mathbf{z}_{m\times 1}, \end{align} where $m<n$ and $\mathbf{D}=[\mathbf{I}_m,\mathbf{0}_{m\times (n-m)}]$ in which $\mathbf{I}_m$ is the identity matrix of size $m$.

Then, what is the following mutual information maximization \begin{align} \max_{p(\mathbf{x}): \mathbb{E}[\mathbf{x}^\mathrm{H}\mathbf{x}]=1} I(\mathbf{x};\mathbf{y}), \end{align} where $(.)^\mathrm{H}$ is the Hermitian operator.

My Guess is that the compex Gaussian distribution is the maximizing distribution because it is unitary invariant.