Finally, adding appropriate multiples of row "t", it can be achieved that all entries in column "j""t" except for that at position ("t","j""t") are zero. This can be achieved by left-multiplication with an appropriate matrix. However, to make the matrix fully diagonal we need to eliminate nonzero entries on the row of position ("t","j""t") as well. This can be achieved by repeating the steps in Step II for columns instead of rows, and using multiplication on the right by the transpose of the obtained matrix "L". In general this will result in the zero entries from the prior application of Step III becoming nonzero again.