This note is partly expositor). Inequalities relating inversion with, respectively, extraction of principal submatriees and the Hadamard product in the two possible orders are developed in a simple and unified way for positive definite matrices. These inequalities are known, hut we also characterize the cases of equality and strict inequality. A by-product is, for example, a pleasant proof of an inequality due to Fiedler. In addition, it is shown that the Hadamard product preserves inequalities in a generalization of Schur's observation. In the process, many tools for dealing with the positive semi-definite partial ordering are exhibited.
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| http://dx.doi.org/10.6028/jres.083.039 | DOI Listing |
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