### Matrix completion and semidefinite matrices [article]

Olaf Dreyer
<span title="2021-12-03">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes the determinant, or as the matrix whose inverse has zeroes in those places that were undetermined in the original matrix. This paper extends these uniqueness results to the case of semidefinite matrices. Because the determinant vanishes for singular matrices,
more &raquo; ... and because the inverse does not exist, we introduce a generalized determinant and use generalized inverses to formulate equivalent characterizations in the semidefinite case. For a class of matrices that are singular but of maximal rank unique characterizations can be given, just as in the positive definite case.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2112.03758v1">arXiv:2112.03758v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zosk467cyjamdbtc4shrcx4mte">fatcat:zosk467cyjamdbtc4shrcx4mte</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20211211093148/https://arxiv.org/pdf/2112.03758v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/08/b9/08b9af3a8a4076bd1b9545b774e7fe69e3f7ba75.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2112.03758v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>