Editing Smith normal form (section)

===Step III: Eliminating entries===
Finally, adding appropriate multiples of row ''t'', it can be achieved that all entries in column ''j''<sub>''t''</sub> except for that at position (''t'',''j''<sub>''t''</sub>) 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''<sub>''t''</sub>) 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.

However, notice that each application of Step II for either rows or columns must continue to reduce the value of <math>\delta(a_{t,j_t})</math>, and so the process must eventually stop after some number of iterations, leading to a matrix where the entry at position (''t'',''j''<sub>''t''</sub>) is the only non-zero entry in both its row and column.

At this point, only the block of ''A'' to the lower right of (''t'',''j''<sub>''t''</sub>) needs to be diagonalized, and conceptually the algorithm can be applied recursively, treating this block as a separate matrix. In other words, we can increment ''t'' by one and go back to Step I.