For a nonsingular square matrix Av, the transformation by Gaussian elimination of Av into the form LDUv, where Lv is a lower triangular matrix, Dv is a diagonal matrix, and Uv is an upper triangular matrix. It can be written so that the diagonal elements of Lv and Uv are equal to one and Dv is the diagonal matrix of pivots. LU matrix decomposition; Matrices and matrix algebra.