A unified understanding of the two formulae for the traces of the inverse powers of a positive definite symmetric tridiagonal matrix

2014/11/13 by Takumi Yamashita, Yamashita, Takumi
Computer Science · Mathematics · Physics and Astronomy · #15A18 #15A99 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1411.3463

openalex publication_date 2014/11/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

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 ) = \textrmTr ( ( B^\top B )^{- M} ) = \textrmTr ( ( B B^\top )^{- M} ) ( M = 1, 2, … ) 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 preceding works in terms of matrix theory is given. Some relationships between some quantities in the preceding works are also given. From these relationships, the new subtraction-free formula for the traces J_M ( B ) is obtained.

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