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

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.