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Given, Toeplitz matrix $T \in R^{n \times n}$: $$ T= \begin{bmatrix} \tau_0 & \tau_1 & \cdots & \tau_{n-1} \\ \tau_1 & \tau_0 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \tau_1 \\ \tau_{n-1} & \cdots & \tau_1 & \tau_0 \\ \end{bmatrix} $$ and denoted the $k$-th order leading principal submatrix of $T$ by $T_k$.

If $T$ is a positive definite matrix, how to prove the following inequality: $$ \rm{det} T_{k+1} \le \frac{(\rm{det} T_k)^2}{\rm{det} T_{k-1}} $$ ,where $\forall k \in \{1, \cdots, n\}$.

And, when the equality is attained?

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  • $\begingroup$ Would the Szegő_limit_theorem be relevant to your question? $\endgroup$ – Vlad May 26 '17 at 1:29
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    $\begingroup$ Have you tried anything? If you have, go ahead and include your work by editing the post. $\endgroup$ – StubbornAtom May 26 '17 at 6:09

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