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.

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.

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.