Successively Ordered Elementary Bidiagonal Factorization
Charles R. Johnson, Dale D. Olesky, P. van den Driessche · SIAM Journal on Matrix Analysis and Applications · 2001
Let D be a diagonal matrix and $E_{ij}$ denote the n-by-n matrix with a 1 in entry $(i,j)$ and 0 in every other entry. An n-by-n matrix A has a successively ordered elementary bidiagonal $(SEB)$ factorization if it can be factored as \begin{equation*} A= \left(\prod_{k=1}^{n-1} \prod_{j=n}^{k+1} L_j(s_{jk})\right) \; D \; \left(\prod_{k=n-1}^{1} \prod_{j=k+1}^{n}U_j(t_{kj})\right), \end{equation*} in which $L_j(s_{jk})=I+s_{jk}E_{j,j-1}$ and $U_j(t_{kj})=I+t_{kj}E_{j-1,j}$ for some scalars $s_{jk},t_{kj}$. Note that some of the parameters $% s_{jk},t_{kj}$ may be zero, and the order of the bidiagonal factors is fixed. If this factorization corresponds to reduction of A to D via successive row/column operations in the specified order, it is called an elimination $SEB$ factorization. New rank conditions are formulated that are proved to be necessary and sufficient for matrix A to have such a factorization. These conditions are related to known but more restrictive properties that ensure a bidiagonal factorization as above, but with all parameters $s_{jk},t_{kj}$ nonzero.