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Row Reduced Form

Module by: Doug Daniels, Steven J. Cox. E-mail the authors

Summary: This module has two goals: to define the elementary row operations for all matrices and to define the 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.

Definition 1: Elementary Row Operations
1. You may swap any two rows.
2. 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
( 0100 -1010 0001 1234 ) 0100 -1010 0001 1234
(1)
it seems wise to swap the first and fourth rows and so arrive at
( 1234 0100 -1010 0001 ) 1234 0100 -1010 0001
(2)
adding the first row to the third now produces
( 1234 0100 0244 0001 ) 1234 0100 0244 0001
(3)
subtracting twice the second row from the third yields
( 1234 0100 0044 0001 ) 1234 0100 0044 0001
(4)
a matrix with zeros below its diagonal. This procedure is not restricted to square matrices. For example, given
( 1111 2442 3553 ) 1111 2442 3553
(5)
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
( 1111 2442 0220 ) 1111 2442 0220
(6)
and then subtract twice the first row from the second,
( 1111 0220 0220 ) 1111 0220 0220
(7)
and finally subtract the second row from the third,
( 1111 0220 0000 ) 1111 0220 0000
(8)
It helps to label the before and after matrices.
Definition 2: The Row Reduced Form
Given the matrix AA we apply elementary row operations until each nonzero below the diagonal is eliminated. We refer to the resulting matrix as A red 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,

Definition 3: Pivot Row
Each nonzero row of A red A red is called a pivot row.
Definition 4: Pivot
The first nonzero term in each row of A red A red is called a pivot.
Definition 5: Pivot Column
Each column of A red A red that contains a pivot is called a pivot column.
Definition 6: Rank
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.

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

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Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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