`We exploit the fact that row transformations do not change the
solution of the system and transform the system succesively
as follows.
`

`In matrix notation, multiplying
the first row by and subtracting
it from the second row leads to
`

`We store the multiplier as follows. Take a matrix ;
since the multiplier is used to
cancel , we store it into .
`

`Multiplying the first row by and subtracting it from the third gives
`

`Finally, by multiplying the second row by and subtracting it from the third
we obtain
`

(14.6) |

`Now, all the elements in the sub-diagonal positions of are zero,
and only the elements above the main diagonal are left.
This transformed is in ``upper triangular form'';
we usually denote it by . In the same time, the matrix of multipliers
has only zero elements above the main diagonal; it is in
``lower triangular form'' ( stands for lower). Note that all
the diagonal elements of are , while the diagonal elements of
can take any values, without any restriction.
`

`If we multiply and we obtain the original matrix
`

(14.7) |

`The relation is called the ``LU decomposition'' of A.
Since has zeros below the main diagonal, we can use this space to store
the elements of ; therefore, we can represent the LU decomposition
compactly as
`