Elementary bidiagonal factorizations Original Research Article

An elementary bidiagonal (EB) matrix has every main diagonal entry equal to 1, and exactly one off-diagonal nonzero entry that is either on the sub- or super-diagonal. If matrix A can be written as a product of EB matrices and at most one diagonal matrix, then this product is an EB factorization of A. Every matrix is shown to have an EB factorization, and this is related to LU factorization and Neville elimination. The minimum number of EB factors needed for various classes of n-by-n matrices is considered. Some exact values for low dimensions and some bounds for general n are proved; improved bounds are conjectured. Generic factorizations that correspond to different orderings of the EB factors are briefly considered.