3.9: Supplements - Row Reduced Form
Row Reduction
A central goal of science and engineering is to reduce the complexity of a model without sacrificing its integrity. Applied to matrices, this goal suggests that we attempt to eliminate nonzero elements and so 'uncouple' the rows. In order to retain its integrity the elimination must obey two simple rules.
- You may swap any two rows.
- You may add to a row a constant multiple of another row.
With these two elementary operations one can systematically eliminate all nonzeros below the diagonal. For example, given
\[\begin{pmatrix} {0}&{1}&{0}&{0}\\ {-1}&{0}&{1}&{0}\\ {0}&{0}&{0}&{1}\\ {1}&{2}&{3}&{4} \end{pmatrix} \nonumber\]
it seems wise to swap the first and fourth rows and so arrive at
\[\begin{pmatrix} {1}&{2}&{3}&{4}\\ {0}&{1}&{0}&{0}\\ {-1}&{0}&{1}&{0}\\ {0}&{0}&{0}&{1} \end{pmatrix} \nonumber\]
adding the first row to the third now produces
\[\begin{pmatrix} {1}&{2}&{3}&{4}\\ {0}&{1}&{0}&{0}\\ {0}&{2}&{4}&{4}\\ {0}&{0}&{0}&{1} \end{pmatrix} \nonumber\]
subtracting twice the second row from the third yields
\[\begin{pmatrix} {1}&{2}&{3}&{4}\\ {0}&{1}&{0}&{0}\\ {0}&{0}&{4}&{4}\\ {0}&{0}&{0}&{1} \end{pmatrix} \nonumber\]
a matrix with zeros below its diagonal. This procedure is not restricted to square matrices. For example, given
\[\begin{pmatrix} {1}&{1}&{1}&{1}\\ {2}&{4}&{4}&{2}\\ {3}&{5}&{5}&{3} \end{pmatrix} \nonumber\]
we start at the bottom left then move up and right. Namely, we subtract 3 times the first row from the third and arrive at
\[\begin{pmatrix} {1}&{1}&{1}&{1}\\ {2}&{4}&{4}&{2}\\ {0}&{2}&{2}&{0} \end{pmatrix} \nonumber\]
and then subtract twice the first row from the second,
\[\begin{pmatrix} {1}&{1}&{1}&{1}\\ {0}&{2}&{2}&{0}\\ {0}&{2}&{2}&{0} \end{pmatrix} \nonumber\]
and finally subtract the second row from the third,
\[\begin{pmatrix} {1}&{1}&{1}&{1}\\ {0}&{2}&{2}&{0}\\ {0}&{0}&{0}&{0} \end{pmatrix} \nonumber\]
It helps to label the before and after matrices.
Given the matrix \(A\) we apply elementary row operations until each nonzero below the diagonal is eliminated. We refer to the resulting matrix as \(A_{red}\).
Uniqueness and Pivots
As there is a certain amount of flexibility in how one carries out the reduction it must be admitted that the reduced form is not unique. That is, two people may begin with the same matrix yet arrive at different reduced forms. The differences however are minor, for both will have the same number of nonzero rows and the nonzeros along the diagonal will follow the same pattern. We capture this pattern with the following suite of definitions,
Each nonzero row of \(A_{red}\) is called a pivot row.
The first nonzero term in each row of \(A_{red}\) is called a pivot .
Each column of \(A_{red}\) that contains a pivot is called a pivot column.
The number of pivots in a matrix is called the rank of that matrix.
Regarding our example matrices, the first has rank 4 and the second has rank 2.
Row Reduction in MATLAB
MATLAB's
rref
command goes full-tilt and attempts to eliminate ALL off diagonal terms and to leave nothing but ones on the diagonal. I recommend you try it on our two examples. You can watch its individual decisions by using
rrefmovie
instead.