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Lately I've been dealing with positive definite matrices in my research (in the context of them being covariance matrices), and, am wondering if anyone knows of a comprehensive list of inequalities involving determinants and minors of these matrices.

I have Horn and Johnson's "Matrix Analysis" which has proofs of the basic inequalities in Chapter 7.8 (s.t. Hadamard's, Fischer's, Szasz's, Minkowski's, and others), and found this article by the esteemed duo of Cover and Thomas to be helpful (they prove the above inequalities using information theoretic tools, and provide a few other ones derived using that machinery).

Unless the above is it (which I doubt), a comprehensive list would be very useful. Can anyone help?

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It's been a while, but maybe you find this hint useful nonetheless: since you are also interested in minors you possibly may want to have a look at Trudinger and Lin's paper 'On some inequalities for elementary symmetric functions', Bull. Austral. Math. Soc. Vol. 50 (1994) [317-326].

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