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

TL;DR

This paper unifies and clarifies existing formulas for computing traces of inverse powers of positive definite symmetric tridiagonal matrices, introducing a new subtraction-free formula and exploring their properties and relationships.

Contribution

It presents a new subtraction-free formula for these traces and clarifies properties and relationships of existing formulas, enhancing understanding and computation.

Findings

New subtraction-free formula for traces $J_M(B)$

Relationships between previous formulas are established

Properties of formulas are clarified and interpreted in matrix theory

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 $J_{M} ( B ) = \textrm{Tr} ( ( B^{\top} B )^{- M} ) = \textrm{Tr} ( ( B B^{\top} )^{- M} )$ $( M = 1, 2, \dots )$ 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 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.