This might be a simple problem for some. Let there be a real positive semidefinite Toeplitz matrix $\mathbf{R}_m$ of dimension $m+1$. Denoting its diagonal elements as $a_0$, we have Tr$(\mathbf{R}_m)=(m+1)a_0$. Moreover, as $\mathbf{R}_m$ is symmetric, it has the block form \begin{align} \mathbf{R}_m=\begin{bmatrix} \mathbf{R}_{m-1} & \mathbf{r}_m \\ \mathbf{r}^\mathsf{T}_m & a_0 \end{bmatrix}. \end{align} Clearly $\text{Tr}(\mathbf{R}_m)=\frac{m+1}{m}\text{Tr}(\mathbf{R}_{m-1})$. However, I want to prove if $\text{Tr}(\mathbf{R}^{-1}_m)\geq\frac{m+1}{m}\text{Tr}(\mathbf{R}^{-1}_{m-1})$. The inequality holds for several sample test cases. But does this extend to all possible $\mathbf{R}_m$?
My approach to the problem was to use the fact \begin{align} \text{Tr}(\mathbf{R}^{-1}_m)&=\text{Tr}(\mathbf{R}^{-1}_{m-1})+\frac{\text{Tr}(\mathbf{R}^{-1}_{m-1}\mathbf{r}_m\mathbf{r}_m^\mathsf{T}\mathbf{R}^{-1}_{m-1})+1}{\kappa} \\ &=\text{Tr}(\mathbf{R}^{-1}_{m-1})+\frac{\mathbf{r}_m^\mathsf{T}\mathbf{R}^{-1}_{m-1}\mathbf{R}^{-1}_{m-1}\mathbf{r}_m+1}{\kappa}, \end{align} where $\kappa$ is the Schur's complement of $\mathbf{R}_{m-1}$, i.e., $\kappa=a_0-\mathbf{r}_m^\mathsf{T}\mathbf{R}^{-1}_{m-1}\mathbf{r}_m$. The above expression stems from the Cholesky decomposition of the block form of $\mathbf{R}_m$ and subsequent inversion. The desired objective is then equivalent to proving if \begin{align} \frac{\mathbf{r}_m^\mathsf{T}\mathbf{R}^{-1}_{m-1}\mathbf{R}^{-1}_{m-1}\mathbf{r}_m+1}{\kappa}\geq\frac{\text{Tr}(\mathbf{R}^{-1}_{m-1})}{m}. \end{align}
An equality that might be useful is $\det(\mathbf{R}_m)=\det(\mathbf{R}_{m-1})\det(\kappa)\Rightarrow\kappa=\frac{\det(\mathbf{R}_m)}{\det(\mathbf{R}_{m-1})}$.