# Orthogonal projections of real positive definite matrices and their determinants

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. $$I \subseteq J$$.

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.