Echelon Form vs Reduced Echelon Form
The matrix obtained after performing several steps of the Gaussian elimination process is said to be in the echelon form or row-echelon form.
A matrix in the echelon form has the following properties.
• All the rows complete with zeros are at the bottom
• The first nonzero values in the nonzero rows shift to the right relative to the first nonzero term in the previous row (see example)
• Any nonzero row starts with 1
Following matrices are in the echelon form:
Continuing the elimination process gives a matrix with all the other terms of a column containing a 1 is zero. A matrix in that form is said to be in the reduced row echelon form.
But the above condition restricts the possibility of having columns with values except 1 and zero. For example, the following is also in the reduced row echelon form.
The reduced row echelon form is found when solving a linear system of equation using Gaussian elimination. The coefficient matrix of the matrix yields the reduced row echelon form and the solution/values for each individual can be easily obtained from a simple computation.
What is the difference between Echelon and Reduced Echelon Form?
• Row echelon form is one format of a matrix obtained by Gaussian elimination process.
• In Row echelon form, the non-zero elements are at the upper right corner, and every nonzero row has a 1. First nonzero element in the nonzero rows shifts to the right after each row.
• Further process of Gaussian elimination gives an even more simplified matrix, where all the other elements in a column containing 1 are zero. A matrix in that form is said to be in reduced row echelon form. That is, in reduced row echelon form, there can be no column that includes 1 and a value other than zero.