Row echelon form: Difference between revisions

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For [[square matrices]], an [[upper triangular matrix]] with nonzero entries on the diagonal is in row echelon form, and a matrix in row echelon form is (weakly) upper triangular. Thus, the row echelon form can be viewed as a generalization of upper triangular form for rectangular matrices.

A matrix is in '''reduced row echelon form''' if it is in row echelon form, the first nonzero entry of each row is equal to <math>1</math> and the ones abovebelow it within the same column equal <math>0</math>. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it. The variant of [[Gaussian elimination]] that transforms a matrix to reduced row echelon form is sometimes called [[Gauss–Jordan elimination]].

A matrix is in '''column echelon form''' if its [[transpose]] is in row echelon form. Since all properties of column echelon forms can therefore immediately be deduced from the corresponding properties of row echelon forms, ''only row echelon forms are considered in the remainder of the article.''