Let $\mathbf{T}$ be the translation operator/matrix in discrete-time domain which can be written as $\mathbf{T} = \mathbf{\Phi} \mathbf{P} \mathbf{\Phi}^*$ where $\mathbf{P} = \exp(-i Diag([w_0, w_1, \ldots, w_M]))$ is a diagonal matrix where $w_k= 2\pi(k-1)/M$. We know that the stochastic signal $\mathbf{x}$ is called wide-sense stationary (WSS) w.r.t. the translation operator on in discrete-time domain $\mathbf{T}$ if and only if for all $r$:
${\mathbf R}_{\mathbf{x}} = \mathbb{E}[\mathbf{x}\mathbf{x}^{*}] = \mathbb{E}\big[(\mathbf{T}^{r} \mathbf{x})\big(\mathbf{T}^{r} \mathbf{x}\big)^{*}\big].$
We need to show that: a process in second-order moment is WSS if and only if \begin{align} {\mathbf R}_{{\mathbf{x}}} = \mathbf{\Phi} {\mathbf S}_{{\mathbf{x}}} \mathbf{\Phi}^{*}, \end{align} where $\mathbf{S}_{\mathbf x}$ is a diagonal matrix with non-negative entries on its main diagonal and $\mathbf{\Phi}^{*}$ is the DFT matrix.
Proof: We can write for the power spectral density matrix as \begin{equation} {\mathbf S}_{{\mathbf{x}}} = \mathbb{E}[\widehat{\mathbf{x}}\widehat{\mathbf{x}}^{*}] = \mathbf{\Phi}^{*}\mathbb{E}[{\mathbf{x}}{\mathbf{x}}^{*}]\mathbf{\Phi} = \mathbf{\Phi}^{*}{\mathbf R}_{{\mathbf{x}}}\mathbf{\Phi}, \end{equation} and hence $ {\mathbf R}_{{\mathbf{x}}} = \mathbf{\Phi} {\mathbf S}_{{\mathbf{x}}} \mathbf{\Phi}^{*}. $ Via the condition above for WSS signal, one can obtain \begin{equation} \begin{split} {\mathbf R}_{\mathbf{x}} = \mathbb{E}[{\mathbf{x}}{\mathbf{x}}^{*}] &= \mathbb{E}\big[\big({\mathbf{T}^{r}\mathbf{x}}\big) \big(\mathbf{T}^{r}{\mathbf{x}}\big)^{*}\big] \\ &= \mathbf{T}^{r} \mathbb{E}[{\mathbf{x}}{\mathbf{x}}^{*}] \mathbf{T}^{{-r}}. \end{split} \end{equation} On can easily obtain \begin{align} \begin{split} {\mathbf S}_{{\mathbf{x}}} &= \big(\mathbf{\Phi}^{*} \mathbf{T}^{r} \mathbf{\Phi}\big) {\mathbf S}_{{\mathbf{x}}} \big(\mathbf{\Phi}^{*} \mathbf{T}^{r}\mathbf{\Phi}\big)^{-1} \\ &= \mathbf{P}^{r} {\mathbf S}_{{\mathbf{x}}} \mathbf{P}^{-r}, \end{split} \end{align} I just get confused here. How we can say from this that ${\mathbf S}_{{\mathbf{x}}}$ is diagonal which is a well-known fact in signal processing?