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Consider an $n\times n$ positive definite real matrix $(m_{i, j}) = M\in \mathbb{R}^{n \times n}$. For an indexing set $I \subseteq [n]$, denote with $M_I$ the sub-matrix of $M$, consisting of coefficient $m_{i, j}$ s.t. $i, j \in I$. In other words, $M_I$ is obtained via an orthogonal projection of $M$ onto the sub-space indexed by $I$.

Find a sufficient (possibly necessary) condition on the eigenvalues $\lambda_1, \dots, \lambda_n$ of $M$, by which it holds $\det(M_J) \leq \det (M_I)$ for all indexing sets s.t. $I \subseteq J$.

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A sufficient condition is that all $\lambda_i\le1$. This follows immediately from Cauchy's Interlacing Theorem (see also here). If $I\subset J$ the eigenvalues of $M_I$ can be matched with $|I|$ of those of $M_J$ in such a way that the $M_I$ ones are greater than their $M_J$ partners. Since the unmatched values belonging to $M_J$ are all $\le 1$, the result follows.

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