structural:scholar_edition
LU Crout decomposition

Note that the decomposition obtained through this procedure is a Doolittle decomposition: the main diagonal of L is composed solely of 1 s. If one would proceed by removing elements above the main diagonal by adding multiples of the columns (instead of removing elements below the diagonal by adding multiples of the rows), we would obtain a Crout decomposition, where the main diagonal of U is of 1 s.

Another (equivalent) way of producing a Crout decomposition of a given matrix A is to obtain a Doolittle decomposition of the transpose of A. Indeed, if A T = L 0 U 0 {\textstyle A^{\textsf {T}}=L_{0}U_{0}} is the LU-decomposition obtained through the algorithm presented in this section, then by taking L = U 0 T {\textstyle L=U_{0}^{\textsf {T}}} and U = L 0 T {\textstyle U=L_{0}^{\textsf {T}}}, we have that A = L U {\displaystyle A=LU} is a Crout decomposition.