Successively ordered elementary bidiagonal factorization

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 or dered elementary bidiagonal (SEB) factorization if it can be factored as [GRAPHICS] in which L-j (s(jk)) = I + s(jk) E-j,E-j-1 and U-j (t(kj)) = I + t(kj) E-j- 1,E- 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/co lumn operations in the specified order, it is called an elimination SEB fac torization. New rank conditions are formulated that are proved to be necess ary and sufficient for matrix A to have such a factorization. These conditi ons are related to known but more restrictive properties that ensure a bidi agonal factorization as above, but with all parameters s(jk), t(kj) nonzero .