  Bidiagonal matrices are widespread in numerical linear algebra, not least
because of their use in the standard algorithm for computing the singular value
decomposition and their appearance as LU factors of tridiagonal matrices. We
show that bidiagonal matrices have a number of interesting properties that make
them powerful tools in a variety of problems, especially when they are
multiplied together. We show that the inverse of a product of bidiagonal
matrices is insensitive to small componentwise relative perturbations in the
factors if the factors or their inverses are nonnegative. We derive
componentwise rounding error bounds for the solution of a linear system $Ax =
b$, where $A$ or $A^{-1}$ is a product $B_1 B_2\dots B_k$ of bidiagonal
matrices, showing that strong results are obtained when the $B_i$ are
nonnegative or have a checkerboard sign pattern. We show that given the \fact\
of an $n\times n$ totally nonnegative matrix $A$ into the product of bidiagonal
matrices, $\|A^{-1}\|_{\infty}$ can be computed in $O(n^2)$ flops and that in
floating-point arithmetic the computed result has small relative error, no
matter how large $\|A^{-1}\|_{\infty}$ is. We also show how factorizations
involving bidiagonal matrices of some special matrices, such as the Frank
matrix and the Kac--Murdock--Szeg\"o matrix, yield simple proofs of the total
nonnegativity and other properties of these matrices.
