{"paper":{"title":"A unified understanding of the two formulae for the traces of the inverse powers of a positive definite symmetric tridiagonal matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Takumi Yamashita","submitted_at":"2014-11-13T08:34:11Z","abstract_excerpt":"For an upper bidiagonal matrix $B$ where all the diagonal and the upper subdiagonal entries are positive, two subtraction-free formulae for computation of the traces $J_{M} ( B ) = \\textrm{Tr} ( ( B^{\\top} B )^{- M} ) = \\textrm{Tr} ( ( B B^{\\top} )^{- M} )$ $( M = 1, 2, \\dots )$ have been presented in the two preceding works. A few lower bounds of the minimal singular value of $B$ are obtained from these traces. In this paper, we clarify some properties of these formulae and present a new subtraction-free formula for the traces $J_{M} ( B )$. An interpretation of some quantities in one of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.3463","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}