  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.
