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For an upper bidiagonal matrix $B$ where all the diagonal and the uppersubdiagonal entries are positive, two subtraction-free formulae for computationof 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 twopreceding works. A few lower bounds of the minimal singular value of $B$ areobtained from these traces. In this paper, we clarify some properties of theseformulae 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 termsof matrix theory is given. Some relationships between some quantities in thepreceding works are also given. From these relationships, the newsubtraction-free formula for the traces $J_{M} ( B )$ is obtained.

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