The algorithm, first introduction

We briefly summarize what we did above.

We started with the first equation and the first variable . Since the coefficient of was zero, we swapped the first equation with the second one.

The current active row is 1 and active column is 1.
Now we have a pivot in the M1[1,1] entry. We use it to zero out all entries below it in the current column. This gives M2 .

We are done with the first row and column, but we keep on copying them along. Now the active ones are row 2 and column 2. The new pivot is M2[2,2] . We zero out entries below it to produce M3 .

Next active row/column are 3,3 respectively. The pivot is M3[3,3] and this is the end, since there are no entries below.

The last operation was to make all pivots into , just for convenience for the remaining work. This produces M4 .

This is the work for gausselim , followed by beginning of gaussjord .

Finally, gaussjord , consists of cleaning out all entries above the pivots. We proceed in reverse order to avoid destrying already created zeros.

At the end of this, the equations are essentially solved!

The next remark is someting to think about and experiment with. It will be justified later.

In general, there is one pivot in each row, unless the row becomes zero. All zero rows are always at the bottom. The pivot in each row also belongs to a column and the column numbers march along to the right. There can be a few extra columns left, which correspond to free variables.

In the first part of the algorithm, all entries below pivots are made zero, in the middle (optional) part, all pivots are made , and finally, all entries above pivots are made zero. Thus, if we only keep the pivot columns and corresponding rows, then we see an identity matrix!