row echelon form

A matrix is said to be in row echelon formMathworldPlanetmath if each non-zero row has more leading zeros than the previous row. Row-echelon form is the key idea underlying the Gaussian eliminationMathworldPlanetmath algorithm and LU factorization.

Let us give the precise definition. Let (Mij) be an n×m matrix. For each row i=1,,n define the pivot position Pi to be either the minimum value of j=1,,m for which Mij0, or if the row consists entirely of zeros. A matrix is in echelon form if for all i>1, either Pi= or Pi-1<Pi.

Examples of matrices in row echelon form include,


Note that if a matrix is an echelon form, then necessarily rows which are composed completely of zeros will be grouped at the bottom of the matrix. Also note that if several rows have the same number of leading zeros then the matrix is not in row echelon form unless the rows in question are composed entirely of zeros.

Title row echelon form
Canonical name RowEchelonForm
Date of creation 2013-03-22 12:14:11
Last modified on 2013-03-22 12:14:11
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 12
Author rmilson (146)
Entry type Definition
Classification msc 15A06
Related topic GaussianElimination
Defines echelon form