Additional notes on linear algebra |
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© Leon van Dommelen |
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1 Gaussian Elimination
To do Gaussian (or forward) elimination in this class, the general
procedure consists of four basic steps. Take the initial
submatrix
to look at to be the complete given matrix.
Then:
- GE I
- If there is only one row left in the submatrix currently
being looked at, you are done with Gaussian elimination. And so you
are if there are no more nonzero coefficients in the submatrix.
- GE II
- In the submatrix currently being looked at, identify the
first column that still has a nonzero coefficient. Any
earlier columns no longer appear anywhere, and should be dropped
from the submatrix being looked at.
- GE III
- In the submatrix currently being looked at, use the
first row to eliminate the first coefficient from the
subsequent rows. Normally, you do that by subtracting
suitable multiples of the first row from the subsequent ones. Step
GE III can be described as “creating zeros below the
pivot.” Note that this is only possible if the coefficient in
the first column of the first row, called the pivot, is
nonzero. In general, before doing the actual step GE III, you may
need to exchange the current first row with a later
one that will produce a better pivot. The next section will explain
that in more detail.
- GE IV
- Drop the first row and the first column from the
submatrix being looked at. With the so-reduced submatrix, repeat
the process starting from step GE I.
This process will give rise to a so-called “echelon
form” of the matrix. That is a special case of an upper
triangular (if square) or upper trapezoidal (if not square) matrix.
Additional note: the eliminations in GE II can be done in a single
step for all rows below the pivot. You do not have to show each row
using a separate matrix. You must however explicitly show the
multiplier(s) used.